Evolutionary Integral Equations and Applications

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Modern Birkhäuser Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkhäuser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain accessible to new generations of students, scholars, and researchers.

Jan Prüss

Evolutionary Integral Equations and Applications

Reprint of the 1993 Edition

Jan Prüss Institut für Mathematik Martin-Luther-Universität Halle-Wittenberg Halle Germany

ISBN 978-3-0348-0498-1 ISBN 978-3-0348-0499-8 (eBook) DOI 10.1007/978-3-0348-0499-8 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2012944422

Mathematics Subject Classification (2010): 45N05, 34K30, 47D06, 47D09, 47N20, 76A10, 78A25 © Springer Basel 1993 Reprint of the 1st edition 1993 by Birkhäuser Verlag, Switzerland Originally published as volume 87 in the Monographs in Mathematics series This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Cover design: deblik, Berlin Printed on acid-free paper

Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . 1 Evolutionary Integral Equations 2 Model Problems . . . . . . . . . 3 Preliminary Discussions . . . . . 4 Equations of Scalar Type . . . . 5 Nonscalar Problems . . . . . . . 6 Equations on the Line . . . . . . 0

I

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Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 Some Notation . . . . . . . . . . . . . . . . . . . . . . . . 0.2 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 0.3 Volterra Equations with Bounded Kernels . . . . . . . . . 0.4 Convolution Equations with Bounded Kernels on the Line 0.5 The Spectrum of Functions of Subexponential Growth . . 0.6 The Spectrum of Functions Growing Polynomially . . . .

ix

. xi . xi . xii . xv . xvii . xx . xxiii . . . . . . .

1 1 4 12 15 19 22

Equations of Scalar Type 1

Resolvents . . . . . . . . . . . . . . . . . . . . 1.1 Well-posedness and Resolvents . . . . . 1.2 Inhomogeneous Equations . . . . . . . . 1.3 Necessary Conditions for Well-posedness 1.4 Perturbed Equations . . . . . . . . . . . 1.5 The Generation Theorem . . . . . . . . 1.6 Integral Resolvents . . . . . . . . . . . . 1.7 Comments . . . . . . . . . . . . . . . . .

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30 30 33 36 39 42 45 47

2

Analytic Resolvents . . . . . . . . . . . 2.1 Definition and First Properties . . 2.2 Generation of Analytic Resolvents 2.3 Examples . . . . . . . . . . . . . . 2.4 Spatial Regularity . . . . . . . . .

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49 49 52 55 56

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v

vi

Contents 2.5 2.6 2.7

Perturbed Equations . . . . . . . . . . . . . . . . . . . . . . Maximal Regularity . . . . . . . . . . . . . . . . . . . . . . Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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60 62 64

3

Parabolic Equations . . . . . . . . . . . 3.1 Parabolicity . . . . . . . . . . . . . 3.2 Regular Kernels . . . . . . . . . . 3.3 Resolvents for Parabolic Equations 3.4 Perturbations . . . . . . . . . . . . 3.5 Maximal Regularity . . . . . . . . 3.6 A Representation Formula . . . . . 3.7 Comments . . . . . . . . . . . . . . Appendix: k-monotone Kernels . . . . . .

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68 68 69 73 76 77 79 81 83

4

Subordination . . . . . . . . . . . . . . . . . . . 4.1 Bernstein Functions . . . . . . . . . . . . . 4.2 Completely Positive Kernels . . . . . . . . . 4.3 The Subordination Principle . . . . . . . . 4.4 Equations with Completely Positive Kernels 4.5 Propagation Functions . . . . . . . . . . . . 4.6 Structure of Subordinated Resolvents . . . 4.7 Comments . . . . . . . . . . . . . . . . . . . Appendix: Some Common Bernstein Functions . .

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90 90 94 99 103 106 110 115 119

5

Linear Viscoelasticity . . . . . . . . . . . . . . . . 5.1 Balance of Momentum and Constitutive Laws 5.2 Material Functions . . . . . . . . . . . . . . . 5.3 Energy Balance and Thermoviscoelasticity . . 5.4 Some One-dimensional Problems . . . . . . . 5.5 Heat Conduction in Materials with Memory . 5.6 Synchronous and Incompressible Materials . . 5.7 A Simple Control Problem . . . . . . . . . . 5.8 Comments . . . . . . . . . . . . . . . . . . . .

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122 122 128 132 135 139 141 146 148

. . . . . . . . . . . . Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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152 152 155 159 162 167 170 175 182

II Nonscalar Equations 6

Hyperbolic Equations of Nonscalar Type . . 6.1 Resolvents of Nonscalar Equations . . . . . 6.2 Well-posedness and Variation of Parameters 6.3 Hyperbolic Perturbation Results . . . . . . 6.4 The Generation Theorem . . . . . . . . . . 6.5 Convergence of Resolvents . . . . . . . . . . 6.6 Kernels of Positive Type in Hilbert spaces . 6.7 Hyperbolic Problems of Variational Type . 6.8 Comments . . . . . . . . . . . . . . . . . . .

Contents

vii

7

Nonscalar Parabolic Equations . . . . . . . . . . . . . . . 7.1 Analytic Resolvents . . . . . . . . . . . . . . . . . . . 7.2 Parabolic Equations . . . . . . . . . . . . . . . . . . . 7.3 Parabolic Problems of Variational Type . . . . . . . . 7.4 Maximal Regularity of Perturbed Parabolic Problems 7.5 Resolvents for Perturbed Parabolic Problems . . . . . 7.6 Uniform Bounds for the Resolvent . . . . . . . . . . . 7.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . .

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185 185 188 190 193 197 201 211

8

Parabolic Problems in Lp -Spaces . . . . . . . 8.1 Operators with Bounded Imaginary Powers 8.2 Vector-Valued Multiplier Theorems . . . . . 8.3 Sums of Commuting Linear Operators . . . 8.4 Volterra Operators in Lp . . . . . . . . . . 8.5 Maximal Regularity in Lp . . . . . . . . . . 8.6 Strong Lp -Stability on the Halfline . . . . . 8.7 Comments . . . . . . . . . . . . . . . . . . .

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212 212 215 217 221 227 229 232

9

Viscoelasticity and Electrodynamics with Memory 9.1 Viscoelastic Beams . . . . . . . . . . . . . . . . . . 9.2 Viscoelastic Plates . . . . . . . . . . . . . . . . . . 9.3 Thermoviscoelasticity: Strong Approach . . . . . . 9.4 Thermoviscoelasticity: Variational Approach . . . 9.5 Electrodynamics with Memory . . . . . . . . . . . 9.6 A Transmission Problem for Media with Memory . 9.7 Comments . . . . . . . . . . . . . . . . . . . . . . .

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236 236 240 242 246 248 251 253

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III Equations on the Line 10

Integrability of Resolvents . . . . . . . . . 10.1 Stability on the Halfline . . . . . . . . 10.2 Parabolic Equations of Scalar Type . 10.3 Subordinated Resolvents . . . . . . . . 10.4 Strong Integrability in Hilbert Spaces 10.5 Nonscalar Parabolic Problems . . . . . 10.6 Comments . . . . . . . . . . . . . . . .

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256 256 260 266 272 277 281

11

Limiting Equations . . . . . . 11.1 Homogeneous Spaces . . . 11.2 Admissibility . . . . . . . 11.3 Λ-Kernels for Compact Λ 11.4 Almost Periodic Solutions 11.5 Nonresonant Problems . . 11.6 Asymptotic Equivalence . 11.7 Comments . . . . . . . . .

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284 284 287 289 293 296 299 303

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viii

Contents 12

Admissibility of Function Spaces . 12.1 Perturbations: Hyperbolic Case . 12.2 Subordinated Equations . . . . . 12.3 Admissibility in Hilbert Spaces . 12.4 Λ-kernels for Parabolic Problems 12.5 Maximal Regularity on the Line 12.6 Perturbations: Parabolic Case . 12.7 Comments . . . . . . . . . . . . .

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306 306 308 310 315 318 320 322

13

Further Applications and Complements . . . . . 13.1 Viscoelastic Timoshenko Beams . . . . . . . . . 13.2 Heat Conduction in Materials with Memory . . 13.3 Electrodynamics with Memory . . . . . . . . . 13.4 Ergodic Theory . . . . . . . . . . . . . . . . . . 13.5 Semilinear Equations . . . . . . . . . . . . . . . 13.6 Semigroup Approaches . . . . . . . . . . . . . . 13.7 Nonlinear Equations with Accretive Operators

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323 323 326 329 333 335 338 342

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Preface During the last two decades the theory of abstract Volterra equations has undergone rapid development. To a large extent this was due to the applications of this theory to problems in mathematical physics, such as viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and to the need of tools to tackle the problems arising in these fields. Many interesting phenomena not found with differential equations but observed in specific examples of Volterra type stimulated research and improved our understanding and knowledge. Although this process is still going on, in particular concerning nonlinear problems, the linear theory has reached a state of maturity. In recent years several good books on Volterra equations have appeared. However, none of them accounts for linear problems in infinite dimensions, and therefore this part of the theory has been available only through the – meanwhile enormous – original literature, so far. The present monograph intends to close this gap. Its aim is a coherent exposition of the state of the art in the linear theory. It brings together and unifies most of the relevant results available at present, and should ease the way through the original literature for anyone intending to work on abstract Volterra equations and its applications. And it exhibits many problems in the linear theory which have not been solved or even not been considered, so far. Even if one is interested in nonlinear problems only, it is useful to know the linear part of the theory; therefore this book offers something to ‘nonlinear’ people, too. A word explaining the title of this monograph seems to be necessary. The equations to be studied are termed ‘abstract Volterra integral equations’ or ‘abstract Volterra integrodifferential equations’ in the literature, depending on their actual form and on the taste of the authors. These terms are somewhat misleading since they refer also to equations containing an abstract Volterra operator, a different notion. The title of this book, however, refers to the evolutionary character of the equations which describe the evolution in time of certain quantities called state of the system. And it refers to the memory present in the equations of state, expressed by certain integrals over a time interval in the past. For these reasons the term ‘evolutionary integral equation’ seems to be more accurate, although ‘Volterra equation’ is used also throughout this book. The ‘Introduction’ contains the formulation of the problems to be studied, some motivating applications, and an outline of the ideas and methods employed in this monograph. There follows a section on notation and on some background material which has been included for the convenience of the reader. The main text consists of 13 sections which are grouped in three chapters. The last section of each chapter contains applications of the developed results to specific equations which are motivated by mathematical physics. The last paragraph of each section is devoted to bibliographical comments and further discussions. A word of caution is necessary concerning the list of references. We have not tried to present a complete bibliography for the subject of this book, but instead to have a representative one, restricting it to the more important contributions. ix

x

Preface

Since it depends very much on personal views and taste what is considered to be essential or important, the selection of the bibliography should by no means expected to be impartial. The numbering of equations, theorems, etc. is evident; for example Theorem 8.2 means Theorem 2 in Section 8, and (10.5) refers to Equation 5 in Section 10. References are indicated by names followed by numbers in square brackets which are found in the bibliography. The number following an item in the ‘Index’ refers to the page where it first appears in the text. Finally, this book would have never been completed without the help of other people. I would like to thank many colleagues for discussions and suggestions, for enduring my smoking habits, and for sending me re- and preprints. Dr. H.J. Warnecke and his crew deserve credit for laser-printing the manuscript several times and for preparing the pictures. Prof. W. Desch, Dr. J. Escher, Dr. R. Schlott read parts of the book, I am very much indebted to them. It is a great pleasure to thank the ‘Heisenberg Referat’ of the ‘Deutsche Forschungsgemeinschaft’, Bonn, for continuous financial support. It is not enough to have an idea for a book but there must also be a publisher; in this respect I would like to thank the editors of this series, in particular Prof. H. Amann for his continuous interest and support, but also Dr. T. Hintermann, Birkh¨ auser Verlag, and collaborators. Most of the credit, however, goes to my family. My wife Bettina TEX-typed the whole manuscript with enthusiasm and carried out the unavoidably many changes and corrections without grumbling. She and also our children S¨ oren and Saskia gave me a lot of encouragement and confidence, and did not complain during the long period of writing this book. Paderborn, September 1992

Jan Pr¨ uss

Introduction This book is devoted to the study of a class of abstract linear integral equations describing the evolution in time of the state of a system, and to applications of the developed theory to problems in mathematical physics. We begin with the description of the class of equations under consideration. 1. Evolutionary Integral Equations Let X and Y be Banach spaces such that Y is densely embedded into X, in symbols d

Y → X, and let B(Y, X) denote the space of bounded linear operators from Y to X, B(X) = B(X, X) for short. Suppose A : ⺢+ → B(Y, X) is measurable and integrable over each compact interval J ⊂ ⺢+ = [0, ∞). Then we consider the equations t

u(t) = 0

A(τ )u(t − τ )dτ + f (t),

t ∈ ⺢+ ,

(1)

t ∈ ⺢,

(2)

on the halfline ⺢+ , and

∞

v(t) = 0

A(τ )v(t − τ )dτ + g(t),

on the line ⺢. Here the functions f and g are given, and we are looking for solutions u(t) of (1) and v(t) of (2) in a sense to be specified later. The family {A(t)}t≥0 of in general unbounded linear operators in the state space X with domain D(A(t)) ≡ Y should be thought of as representing a physical system, u(t) ∈ X resp. v(t) ∈ X as the state of the system at time t, f (t) resp. g(t) as given forces, and (1) resp. (2) as the equations governing the state of the system. In applications, the space X will typically be a space of functions living on a domain Ω ⊂ ⺢n , A(t) an elliptic differential operator and Y the domain of A(t) which may incorporate certain boundary conditions; see 2. below. Apparently, the principle of causality is valid for (1) and (2), since the present state of the system is determined by its history and the present force, but does not depend on the future. Therefore (1) and (2) are evolutionary in character, which explains the term ‘evolutionary integral equations’ for (1) and (2). A second main features of these equations are their time invariance. This means that (1) and (2) do not change their form when a time shift is applied to the solutions. In fact, if u(t) is a solution of (1), then us (t) = u(t + s), s ≥ 0, is again a solution of (1), where f (t) is replaced by s fs (t) = f (t + s) + A(t + τ )u(s − τ )dτ, t ≥ 0. 0

Equation (2) even has the stronger property of translation invariance, i.e. if v(t) is a solution of (2) then the translated function vs (t) = v(t+s), s ∈ ⺢, is a solution of (2) with the translate gs (t) = g(t + s) instead of g(t). These invariance properties mean physically that the system under consideration does not change with time. Mathematically, this makes the powerful tools of transform theory available for xi

xii

Introduction

the study of (1), and (2); more precisely, Laplace transform techniques can be applied to (1) and (2) is amenable to Fourier transform methods. In the finite dimensional case dim X < ∞ (which implies Y = X), (1) and (2) reduce to the well understood ‘linear Volterra integral equations’ on the halfline resp. on the line. This notion has been employed in the literature also for the general unbounded case Y = X; we follow this ‘tradition’ and use the term ‘Volterra equation’ for (1) and (2) everywhere in this book, for the sake of brevity. We are neither concerned here with dim X < ∞ nor with the somewhat more general, but still simple case Y = X, dim X = ∞. However, since they are needed as a tool, in Sections 0.3 and 0.4 the main results for Y = X are reproduced, namely the unique solvability of (1) on each compact interval J = [0, T ] in any of the usual function spaces, and the Paley-Wiener lemma on global solvability of (1) and (2). For an account of the theory of Volterra equations in finite dimensions -linear and also nonlinear- we refer to the recent monograph of Gripenberg, Londen and Staffans [156]. It is time now to present some model problems to explain what kind of concrete equations we have in mind. These problems are described in much greater detail in the sections containing applications, i.e. Sections 5, 9, and 13, where physical interpretations and some background information as well as detailed mathematical treatments can be found. 2. Model Problems A rich source of problems leading to the Volterra equations (1) and (2) is provided by the theory of viscoelastic material behaviour. We begin with some typical examples from this field. (i) Simple Shear The following initial-boundary value problem is a typical example of one-dimensional problems in viscoelasticity, like simple shearing motions, torsion of a rod, simple tension; see Section 5.4. ut (t, x) = u(t, 0) u(0, x)

t 0

da(τ )uxx (t − τ, x) + h(t, x),

= u(t, 1) = 0, t ≥ 0, = u0 (x), x ∈ [0, 1].

t ≥ 0, x ∈ [0, 1], (3)

Here a : ⺢+ → ⺢ is a function of bounded variation on each compact interval J = [0, T ] with a(0) = 0, and the subscripts t or x mean partial differentiation w.r.t. the corresponding variable. To obtain a formulation as an abstract evolutionary integral equation (1), choose a function space, say X = C0 [0, 1], define an operator A0 by means of A0 w(x) = wxx (x) with domain Y = D(A0 ) = {w ∈ X : wxx ∈ X}, and let A(t) = a(t)A0 . If X and Y are equipped with their natural norms, after an

Introduction

xiii

integration w.r.t. time t, (3) becomes (1) with force t f (t) = u0 (·) + h(τ, ·)dτ, t ≥ 0, 0

and state u(t) = u(t, ·). This way, boundary conditions other than those of Dirichlet type can be treated, as well, and of course there are other choices for the state space X. 2

(ii) Viscoelastic Fluids A model for the dynamic behaviour of the velocity field u(t, x) of a ‘linear’ homogeneous isotropic incompressible viscoelastic fluid confined to a domain Ω ⊂ ⺢3 and subject to an external force h(t, x) is the boundary value problem (see Section 5.6) ∞ ut (t, x) = da(τ )Δu(t − τ, x) − ∇p(t, x) + h(t, x), t ∈ ⺢, x ∈ Ω, 0

div u(t, x) = 0, u(t, x) = 0,

t ∈ ⺢, x ∈ Ω,

(4)

t ∈ ⺢, x ∈ ∂Ω,

where p(t, x) denotes the also unknown pressure and the scalar function a(t) is of bounded variation on ⺢+ . Observe that for a(t) ≡ a0 > 0, (4) reduces to the linear Navier-Stokes equation for a Newtonian fluid. Let Pq denote the Helmholtz projection in Lq (Ω)3 to the subspace Lqσ (Ω) of divergence-free Lq -vector fields, and Aq = Pq Δ the Stokes operator; see Section 5.6 for more details. Let Y = D(Aq ) be equipped with the graph norm of Aq and define A(t) = a(t)Aq . The application of Pq to (4) yields (2) in differentiated form, i.e. ∞ u(t) ˙ = dA(τ )u(t − τ ) + g(t), t ∈ ⺢, (5) 0

where g(t) = Pq h(t, ·), and the dot -as always in this book- indicates differentiation w.r.t. time t. 2

(iii) Viscoelastic Beams The viscoelastic version of the Timoshenko beam model, where one end of the beam is clamped while the other is free, reads as follows (cp. Section 9.1). t wtt (t, x) = da(τ )[wtxx (t − τ, x) + φtx (t − τ, x)] + fs (t, x), 0 t φtt (t, x) = de(τ )φtxx (t − τ, x) 0 t −γ da(τ )[wtx (t − τ, x) + φt (t − τ, x)] + fb (t, x), (6) 0

w(t, 0) w(0, x) wt (0, x)

= φ(t, 0) = φx (t, 1) = wx (t, 1) + φ(t, 1) = 0, = w0 (x), = w1 (x),

φ(0, x) = φ0 (x), φt (0, x) = φ1 (x),

xiv

Introduction

where t ≥ 0, x ∈ [0, 1], and fs , fb , w0 , w1 , φ0 , φ1 are given. The functions a(t), e(t) are real valued and of bounded variation on each compact interval J = [0, T ], and γ > 0 is a constant; see Section 9.1 for the physical interpretation of the variables w and φ, of a and e, as well as of the dynamic equations. Choose X = L2 (0, 1) × L2 (0, 1) and define the operator family {A(t)}t≥0 by a(t)∂x2 , a(t)∂x A(t) = (7) −γa(t)∂x , e(t)∂x2 − γa(t) where D(A(t)) ≡ Y = {(w, φ) ∈ W 2,2 (0, 1) : w(0) = φ(0) = φx (1) = wx (1) + φ(1) = 0}. After two integrations w.r.t. time, (6) then takes on the form (1). 2

The classes of evolutionary integral equations studied in this book typically arise in mathematical physics by some constitutive laws of the memory type when combined with the usual conservation laws. The next examples illustrate this point. (iv) Heat Conduction with Memory Let ε(t, x) denote the density of internal energy, θ(t, x) the temperature, q(t, x) the heat flux vector field in a rigid body Ω ⊂ ⺢3 , and let r(t, x) be the external heat supply. Balance of energy then reads as εt (t, x) + div q(t, x) = r(t, x),

t ∈ ⺢, x ∈ Ω,

(8)

and boundary conditions are either prescribed temperature or prescribed heat flux through the boundary, i.e. ◦

θ(t, x) = θb (t, x) on Γb ,

◦

−q(t, x)n(x) = qf (x) on Γf ,

(9)

where n(x) denotes the outer normal of Ω at x ∈ ∂Ω; Γb , Γf ⊂ ∂Ω are closed, ◦

◦

◦

◦

Γb ∩ Γf = ∅, Γb = Γb , Γf = Γf , and Γb ∪ Γf = ∂Ω. For isotropic and homogeneous bodies the general linear time invariant constitutive laws are given by ∞ ε(t, x) = dm(τ )θ(t − τ, x) + ε∞ , (10) 0 ∞ q(t, x) = − dc(τ )∇θ(t − τ, x), t ∈ ⺢, x ∈ Ω; 0

the scalar-valued kernels m(t) and c(t) reflect the properties of the material, but are at least of bounded variation on ⺢+ . Note that the case dm = m0 δ0 , dc = c0 δ0 , δ0 Dirac’s distribution, corresponds to the classical Fourier laws of heat conduction. Thus combining (8), (9), and (10) we obtain the boundary value problem ∞ ∞ dm(τ )θt (t − τ, x) = dc(τ )Δθ(t − τ, x) + r(t, x), t ∈ ⺢, x ∈ Ω, 0

0

θ(t, x)

∞

dc(τ ) 0

∂θ (t − τ, x) ∂n

= θb (t, x), = qf (t, x),

◦

t ∈ ⺢, x ∈Γb , ◦

t ∈ ⺢, x ∈Γf .

(11)

Introduction

xv

There are several ways to convert this problem into a Volterra equation (2) on the line. For more background and a thorough mathematical study of (11) see Sections 5.3, 5.5, and 13.2. 2

(v) Electrodynamics with Memory The situation is similar in linear electrodynamics with memory. The basic Maxwell equations Bt + curl E = 0,

div B = 0,

Dt − curl H + J = 0,

div D = ρ,

(12)

connecting electric field E, magnetic field H, magnetic induction B, electric induction D, free current J , and free charge ρ, have to be supplemented by constitutive equations. For homogeneous and isotropic materials these are given by ∞ ∞ B(t, x) = dμ(τ )H(t − τ, x), D(t, x) = dε(τ )E(t − τ, x), 0 0 ∞ J (t, x) = dσ(τ )E(t − τ, x), x ∈ Ω, t ∈ ⺢. (13) 0

The material functions μ(t), ε(t), σ(t) reflect the properties of the medium under consideration; in the classical case these are simply constant, i.e. B = μH, D = εE, J = σE. Electrodynamics with memory is discussed in Sections 9.5, 9.6, and 13.3. 2

3. Preliminary Discussions In general, the form in which Volterra equations arise in applications is given by Equation (2) on the line, as shown by the examples just given, and (1) arises from (2) as a history value problem; by this we mean the following. Assume v(t) is known for t < 0; then v(t) is a solution of (1), where the forcing term f (t) in (1) contains g(t) as well as the forcing effect of the history of v according to ∞ f (t) = g(t) + A(t + τ )v(−τ )dτ, t ≥ 0. (14) 0

In particular (2) reduces to (1) in case the system is virgin at t = 0, i.e. g(t) ≡ v(t) ≡ 0 for t < 0; this means that up to time t = 0, the system has been in equilibrium and has not been subject to any forces. It might be questioned whether it makes sense to consider (2) for all times since every man-made material or system has been virgin at some time. However, in this respect the history value problem is even more questionable since it is just impossible to record the complete history of a system and to keep this memory for all time. So in this line of arguments the only reasonable problem is that of systems, virgin at t = 0, in which case (1) and (2) coincide. But when considering problems with periodic or almost periodic forcing, the equation to consider is (2), since (1) is only time invariant but not translation invariant, only (2) enjoys the

xvi

Introduction

latter property. In this context the important question arises whether the solutions u(t) of (1) and v(t) of (2) are asymptotic to each other, i.e. whether u(t)−v(t) → 0 as t → ∞, whenever f (t) − g(t) → 0 as t → ∞. Under reasonable assumptions this turns out to be the case, and therefore the term limiting equation of (1) makes sense for (2); the solution v(t) of (2) describes the limiting behaviour of the solution u(t) of (1) as t → ∞, provided f˙ and g˙ are asymptotic to each other. As the above examples show, in many situations the operator family {A(t)}t≥0 turns out to be of the special form A(t) = a(t)A, where a ∈ L1loc (⺢) is a scalarvalued kernel, A a closed linear densely defined operator in X, and Y = D(A) equipped with the graph norm of A. In this case (1) and (2) are said to be of scalar type. The theory is considerably simplified for such problems, also much more complete. It contains two important special cases which are obtained as follows. (i) Choose a(t) ≡ 1; differentiation of (1) yields the abstract Cauchy problem (of first order) u(t) ˙ = Au(t) + h(t), t ∈ ⺢+ , u(0) = u0 , (15) where u0 = f (0), h(t) = f˙(t), and the ‘dot’ indicates differentiation w.r.t. time t. Initial value problem (15) has been studied extensively through the last 60 years, and there are many monographs on the subject semigroup theory dealing with such abstract Cauchy problems. (ii) Choose a(t) ≡ t; differentiating (1) twice results in the second order problem u ¨(t) = Au(t) + h(t),

˙ = u1 , t ∈ ⺢+ , u(0) = u0 , u(0)

(16)

where u0 = f (0), u1 = f˙(0), and h(t) = f¨(t). To some extent it is possible to rewrite (16) as a first order Cauchy problem, i.e. (16) is subsumed by semigroup theory, but there is also an independent theory for (16) which leads to the concept of a cosine family. Similar to the case of second order Cauchy problems (16) attempts have been made to reformulate (1) (and also (2)) as an abstract Cauchy problem (15) in certain spaces of history or forcing functions. In finite dimensions this approach works quite well, although it has the disadvantage of transforming a finite-dimensional problem into an infinite dimensional one. In the infinite dimensional unbounded case, i.e. Y = X, however, it does not work so well and is problematic in several aspects, some of which have been discussed above. The advantage of this approach is of course that the well established theory of semigroups can be employed; this way Volterra equations become a chapter of semigroup theory. But this is not the whole story. Already when reformulating (16) as a first order problem there arises the question of the proper phase space where (u, u) ˙ should live in. This question can only be resolved after (16) has been studied independently: u(t) ∈ X1 , u(t) ˙ ∈ X, where X1 denotes the set of all x ∈ X such that Co(·)x is a C 1 -function; here Co(t) means the cosine family corresponding to

Introduction

xvii

(16). For Volterra equations (1) or (2) this problem becomes much worse; even after an independent study of (1) resp. (2) there is no unique or natural choice of a phase space e.g. of history or forcing functions such that (1) resp. (2) reduce to an abstract Cauchy problem (15). Moreover, such reformulations in general change the nature of the original problem; although (1) resp. (2) may be of parabolic type, the resulting reformulation will always be of hyperbolic type. For these reasons, we study (1) and (2) as they stand. The theory developed thus forms an extension of semigroup theory, and when specialized to the case A(t) ≡ A, reduces to well-known results on semigroups; similarly, for A(t) = tA results on cosine families will be recovered. On the other hand, (1) and (2) are treated very much in the spirit of semigroup theory, and it will be helpful but not neccessary for the reader to be familiar with its basics. Concerning the theory of C0 -semigroups we refer to the monographs of Hille and Phillips [180], Krein [200], Tanabe [319], Davies [77], Fattorini [113], Pazy [267], Goldstein [133], Nagel [255], and Clément, Heijmans, Angenent, van Duijn, de Pagter [47]. Nevertheless, we discuss semigroup methods briefly in Section 13 and give a list of references which is far from being complete. There our point of view is underlined, we show how semigroups in certain phase spaces may be constructed after the underlying Volterra equation has been solved. Nonautonomous problems are also of interest in theory as well as in practice; they occur e.g. as linearizations of nonlinear equations at nonconstant solutions, and arise in the theory of aging of materials with memory. However, due to space considerations and since no ideas other than those presented in this book are known for their treatment (essentially perturbation arguments), the emphasis is put here on autonomous problems for which the theory is much richer. The nonautonomous case is only discussed in some of the comment sections. For understanding this book, only moderate knowledge in Real and Complex Analysis, Functional Analysis, Differential and Integral Equations, and Harmonic Analysis, especially Transform Theory is needed. For convenience of the reader we have included the preliminary Section 0 which contains several results, scattered in the literature, on the vector-valued Laplace transform, on Volterra equations (1) and (2) in the bounded case, and on spectral theory of vector-valued functions. 4. Equations of Scalar Type Having said something about the problems to be studied and being motivated by examples from Mathematical Physics, we want to present now some details on the ideas, methods, and contents of this book. Let us begin with equations of the form (1) which are of scalar type A(t) = a(t)A, i.e. consider

t

u(t) = 0

a(τ )Au(t − τ )dτ + f (t),

t ∈ ⺢+ ,

(17)

where A is a closed linear, densely defined operator in X, and a ∈ L1loc (⺢+ ) a scalar kernel. The study of this class of Volterra equations is the main theme of Chapter 1. A continuous function u : ⺢+ → X is called a strong solution of (17) if

xviii

Introduction

u(t) ∈ D(A) for all t ∈ ⺢+ , Au(t) is continuous, and (17) holds on ⺢+ ; u is called a mild solution of (17) if the convolution (a ∗ u)(t) =

t 0

a(τ )u(t − τ )dτ,

t ∈ ⺢+ ,

belongs to D(A) for all t ∈ ⺢+ , and u(t) = A(a∗u)(t)+f (t) holds on ⺢+ . The basic concept concerning (17) is that of well-posedness which is the direct extension of the corresponding notion usually employed for the abstract Cauchy problem (15). In Section 1 it is shown that well-posedness is equivalent to the existence of a resolvent {S(t)}t≥0 ⊂ B(X) for (17), i.e. a strongly continuous family of bounded linear operators in X which commutes with A and satisfies the resolvent equation

t

S(t)x = x + 0

a(t − τ )AS(τ )xdτ,

t ∈ ⺢+ , x ∈ D(A).

(18)

The resolvent is the central object to be studied in the theory of Volterra equations; it corresponds to the semigroup in the special case a(t) ≡ 1, i.e. for (15), respectively to the cosine family for a(t) ≡ t, i.e. for (16). The importance of the resolvent S(t) is shown by the variation of parameters formula valid for mild solutions u(t) of (17). d u(t) = dt

t 0

S(t − τ )f (τ )dτ,

t ∈ ⺢+ .

(19)

This formula is a generalization of the usual variation of parameters formulae for Cauchy problems of first and second order. Section 1 contains the basic theory of Volterra equations of scalar type. The equivalence between well-posedness of (17) and existence of the resolvent is established, the variation of parameters formula (19) is derived and its elementary properties are studied. It is then used to obtain some simple perturbation results. Elementary spectral theory yields conditions necessary for existence of resolvents which in the case of a normal operator A in a Hilbert space are also sufficient. In particular, we obtain a very useful existence theorem for resolvents. Due to the time invariance of (17), Laplace transform methods can be employed. Formally the solution of (17) is represented in the frequency domain by u ˆ(λ) = (I − a ˆ(λ)A)−1 fˆ(λ)

(20)

ˆ and the Laplace transform H(λ) = S(λ) of the resolvent by H(λ) =

1 (I − a ˆ(λ)A)−1 . λ

(21)

This formula leads to a characterization of Hille-Yosida type for existence of an at most exponentially growing resolvent for (17) in terms of properties of the holomorphic operator family H(λ). This result plays a crucial role in later developments.

Introduction

xix

In Section 2 we take up the study of analytic resolvents, i.e. resolvents S(t) which admit analytic continuation to some sector Σ(0, θ) with vertex 0 and opening angle 2θ. Such resolvents can be easily characterized in terms of boundedness properties of H(λ), but require very smooth kernels a(t). It is shown that in analogy to analytic semigroups, AS(t) is bounded for each t > 0, and optimal bounds are derived. However, if A2 S(t) is bounded, say on an interval (t0 −ε, t0 +ε), ε > 0, then a(t) ≡ a0 and S(t) is already an analytic semigroup. Interpreting A as an elliptic differential operator, this rather striking property of Volterra equations means that there is no infinite spatial smoothing, which is in sharp contrast to partial differential equations of parabolic type. Equations (17) which admit an analytic resolvent are special cases of parabolic Volterra equations, which are studied in Section 3. Parabolicity of (17) is defined in terms of existence and boundedness of λH(λ) in the open right halfplane, a condition easy to check in applications. Together with a certain regularity of the involved kernel a(t), called k-regularity, it leads to very simple sufficient conditions for existence and time-regularity of the resolvent. Similar to parabolic partial differential equations, parabolic Volterra equations allow for improved perturbation results and enjoy the property of maximal regularity of type C α , which means that (17) admits a mild solution u ∈ C α (⺢+ ; X), whenever f ∈ C α (⺢+ ; X); here α ∈ (0, 1). Section 4 is devoted to a thorough study of the so-called subordination principle for Volterra equations. This principle is useful for parabolic as well as for hyperbolic problems and has many applications. To explain it briefly, let b ∈ L1loc (⺢+ ) be Laplace transformable and suppose there is an at most exponentially growing resolvent Sb (t) for (17) with a(t) replaced by b(t). Assume that c(t) is a completely positive function, i.e. a function c ∈ L1loc (⺢+ ) such that k(t), defined by ˆ k(λ) = 1/λ2 cˆ(λ),

λ > 0,

(22)

is a creep function, i.e. nonnegative, nondecreasing, and concave. Then there is a ∈ L1loc (⺢+ ) such that a ˆ(λ) = ˆb(1/ˆ c(λ)),

λ > ω,

(23)

where ω is sufficiently large. By the subordination principle, (17) admits a resolvent Sa (t) as well, which is given by the formula ∞ Sa (t) = − Sb (τ )dτ w(t; τ ), t > 0. (24) 0

Here w(t; τ ) is a function of two variables defined only in terms of c(t), called the propagation function, and it admits for nice physical interpretations in special applications. By the structure of w(t; τ ), (24) yields t/κ Sb (τ )v(t; τ )dτ, t > 0, (25) Sa (t) = Sb (t/κ)e−αt/κ + 0

˙ where κ = limt→0+ k(t) and α = − limt→0+ k(t). Note that (25) is a decomposition of the resolvent Sa (t) of the subordinated equation into one part which is the

xx

Introduction

original resolvent Sb propagated with ‘speed’ 1/κ and exponentially damped with ‘attenuation’ α, plus another part which also propagates with speed 1/κ but due to the presence of the integral is more smooth. Thinking of Sb (t) being a cosine family or a hyperbolic semigroup, (25) with κ > 0, α = ∞ allows for the interesting phenomenon of coexistence of a finite propagation speed and smoothing, i.e. absence of wavefronts. This phenomenon is not observed in linear differential equations, it is a memory effect. The first three subsections of Section 5 deal with the formulation of the basic equations of viscoelasticity, heat conduction in materials with memory, and thermoviscoelasticity. It is shown that in this connection completely positive functions as well as creep functions occur naturally. The remaining subsections are then devoted to special problems which lead to Volterra equations (17) of scalar type, and the theory developed so far is applied. 5. Nonscalar Problems In Chapter 2 we study (1) for operator-valued kernels A ∈ L1loc (⺢+ ; B(Y, X)), a situation which is much more complicated than the scalar case. One reason for this is the lack of a ‘natural’ relation between the spaces Y and X, in contrast to the scalar case where Y = XA , the domain D(A) of A equipped with the graph norm of A, is the only reasonable choice. The notion of a strong solution of (1) extends to this setting in an obvious way; a continuous function u : ⺢+ → Y is called a strong solution for (1) with f ∈ C(⺢+ ; X) if (1) holds on ⺢+ . But already ‘mild solution’ must be defined differently; u : ⺢+ → X continuous is called a mild solution of (1) if it is the limit, uniformly on compact intervals, of strong solutions un of (1) with f replaced by fn , and fn → f , uniformly on compact intervals. The concepts well-posedness and resolvent admit natural extensions to the general setting, with the main difference that we have to consider two resolvent equations now, namely t S(t)y = y + A(t − τ )S(τ )ydτ, t ≥ 0, y ∈ Y, (26) 0

and

t

S(t)y = y + 0

S(t − τ )A(τ )ydτ,

t ≥ 0, y ∈ Y.

(27)

Mild solutions are again represented by the variation of parameters formula (19) whenever there is a weak resolvent S(t) for (1), i.e. {S(t)}t≥0 ⊂ B(X) is strongly continuous and the second resolvent equation (27) holds. However, in contrast to the case of problems of scalar type, uniqueness of strong solutions does not seem to imply uniqueness of mild solutions, and therefore ‘wellposedness’ is not equivalent to existence of a resolvent. Also, time regularity of f (t) alone in general does not imply existence of strong solutions, as simple examples show. Another problem arises with the Hille-Yosida type characterization of resolvents of at most exponential growth, which is only valid in spaces with the Radon-Nikodym property. These imperfections require further notions like pseudo-resolvent and a-regularity of a weak resolvent.

Introduction

xxi

Section 6 deals with the general theory for (1). The basic definitions are introduced, the relations between them are discussed, and the variation of parameters formula is studied. The latter then yields in combination with results on Volterra equations with bounded kernels a fairly general perturbation theorem. The Hille-Yosida type characterization of pseudo-resolvents is proved as well as a convergence theorem for (pseudo-)resolvents of the Trotter-Kato type. This basic theory simplifies considerably if A(t) has a main part, i.e. is of the form t A(t) = a(t)A + a(t − τ )dB(τ ), t ≥ 0, (28) 0

where A is a closed linear operator in X with dense domain D(A), Y = XA , a ∈ L1loc (⺢+ ), and B ∈ BVloc (⺢+ ; B(XA , X)), B(0) = B(0+) = 0, where B(0+) = limt→0+ B(t). If X and Y are Hilbert spaces and A ∈ BVloc (⺢+ ; B(Y, X)) is such that −dA is of positive type, i.e. T t Re (dA(τ )u(t − τ ), u(t))X dt ≤ 0 for each u ∈ C(⺢+ ; Y ), T > 0, (29) 0

0

is satisfied, then the energy inequality |u(t)|X ≤ |f (0)|X +

t 0

|f˙(τ )|X dτ,

t > 0,

(30)

1,1 holds for each mild solution u(t) of (1), where f ∈ Wloc (⺢+ ; X). This inequality then leads to existence of a weak resolvent for (1), even of a resolvent in case A(t) has a main part. Results of this type are important in applications since the abstract energy inequality (30) corresponds to the real energy inequality e.g. in viscoelasticity. It generalizes in a natural way the well-known facts, that maccretive operators are the negative generators of semigroups of contractions, and that negative semidefinite selfadjoint operators in a Hilbert space generate uniformly bounded cosine families. The energy inequality (30) can be considerably strengthened if (1) is of variational form in the following sense. Let V and H be Hilbert spaces such that d

V → H, and let ((·, ·)) resp. (·, ·) denote the inner products in V resp. H, || · || ∗ resp. | · | the corresponding norms in V resp. H. Identifying the antidual H of d

∗

∗

H with H, we obtain H → V by duality, and the antiduality < ·, · > of V, V is related to the inner product of H via < v, h >= (v, h) for all v ∈ V , h ∈ H. Let α : ⺢+ × V × V → ⺓ be such that α(t, ·, ·) is a bounded sesquilinear form on V for each t ≥ 0, and consider the problem of variational type t (w, u(t)) + α(t − τ, w, u(τ ))dτ =< w, f (t) >, t ∈ ⺢+ , w ∈ V, (31) 0

∗

where f ∈ C(⺢+ ; V ). Representing the forms α(t, ·, ·) by bounded linear operators ∗ ∗ −A(t) ∈ B(V, V ) it is obvious that (31) can be written as (1) in X = V , with

xxii

Introduction

Y = V . Now, if the form α is coercive in the sense that there is a constant γ > 0 with T T t [ dα(τ, u(t), u(t − τ )]dt ≥ γ|| u(τ )dτ ||2 , (32) 2 Re 0

0

0

for all u ∈ C(⺢+ ; V ) and T > 0, then (30) improves to t t 2 2 2 |u(t)| + γ|| u(τ )dτ || ≤ |f (0)| + 2 Re < u(t), f˙(τ ) > dτ. 0

(33)

0

Based on this strong energy inequality and some regularity of the form α w.r.t. t ≥ 0, existence of a resolvent can be proved, along with several uniform estimates and regularity properties. This result is shown to cover nonisotropic hyperbolic viscoelasticity in Section 9, and also applies to thermoviscoelasticity and several other problems. In Section 7 we extend the results of Sections 2 and 3 on analytic resolvents and parabolic equations to the nonscalar case. This is straightforward and does not lead to further complications; maximal regularity of type C α carries over easily, too. For equations in variational form, coercive estimates of parabolic type lead to especially nice results, as one should expect. We then consider the perturbed problem t t s u(t) = A(t−τ )u(τ )dτ + a(t−s)[ dB(τ )u(s−τ )]ds+f (t), t ∈ ⺢+ , (34) 0

0

0

where the unperturbed equation is parabolic, admits an a-regular resolvent with some additional smoothness properties, and B ∈ BVloc (⺢+ ; B(Y, X)), B(0) = B(0+) = 0. We show that (34) also enjoys the maximal regularity property of type C α and of type Bpα,q as well. This result, together with L∞ -estimates for (34) which are quite delicate to prove, yields a resolvent for the perturbed problem (34). A quite different approach to (1) is taken up in Section 8. In the scalar case A(t) = a(t)A, (1) can be reformulated as Au + Bu = g,

(35)

in spaces of functions on ⺢+ , where the operators A and B are defined by (Au)(t) = −Au(t),

t ∈ ⺢+ ,

(36)

and B formally via Laplace transforms (Bu)(λ) = (1/ˆ a(λ))ˆ u(λ),

Re λ > 0.

(37)

One can then apply the sum method to obtain ‘solutions’ of (35), without going into details here. In case X belongs to the class HT and the problem under consideration is parabolic, this approach works quite well in Lp -spaces and can even be extended to equations of nonscalar type with main part. Since this approach

Introduction

xxiii

is still very much under progress, we do not give all of the proofs required, but only an exposition of the general method, as well as some of its implications for (1) which are available at present. Finally, in Section 9 the discussions of Section 5 are continued. A brief (heuristical) derivation of some common viscoelastic beam models is presented, and the preceding results are applied to study their well-posedness. Models for viscoelastic plates are formulated according to the literature, i.e. the viscoelastic linear Midlin-Timoshenko plate and the linear Kirchhoff model. Concerning three dimensional thermo-viscoelasticity two approaches are persued, one based on results for equations of scalar type plus perturbations, requiring the material relations to be ‘almost separable’, and the second following the variational approach, which relies on Sections 6.7 and 7.3. The final Sections 9.5 and 9.6 deal with the formulation of linear electrodynamics with memory and applications of several of the results, obtained so far. 6. Equations on the Line The third chapter is devoted to the study of the equation on the line (2) and its relations to (1). This connection is particularly simple if (1) admits a resolvent S(t) which is integrable, i.e. |S(t)| ≤ ϕ(t) for a.a. t ≥ 0, ϕ ∈ L1 (⺢+ ) ∩ C0 (⺢+ ). In fact, in this case the solution of (2) is given by ∞ v(t) = S(τ )g(t ˙ − τ )dτ, 0

t ∈ ⺢.

(38)

(39)

From this formula the solvability properties of (2) can easily be read off; moreover, the variation of parameters formula for (1) yields t u(t) − v(t) = S(t)g(0) + S(τ )(f˙(t − τ ) − g(t ˙ − τ ))dτ 0 ∞ − S(τ )g(t ˙ − τ )dτ, t ≥ 0,

(40)

t

hence u(t) − v(t) → 0 as t → ∞, whenever g˙ ∈ Lp (⺢− ; X) for some p ∈ [1, ∞], and f˙(t) − g(t) ˙ → 0. This shows that the solutions u(t) of (1) and v(t) of (2) are asymptotic to each other as t → ∞; this justifies the name limiting equation of (1) for problem (2). Thus the solvability behaviour of (1) and (2) is especially nice in the stable case, when the resolvent for (1) is integrable. For this reason, but also since stability is very important in applications, Section 10 is devoted to the study of integrability of resolvents. The discussion begins with the derivation of necessary conditions for the important special case of equations of scalar type. These conditions are given in terms of the symbol H(λ) of (17) defined by (21), and basically require existence and boundedness of H(λ) on the closed right halfplane λ ∈ ⺓+ . Un-

xxiv

Introduction

fortunately, these conditions are in general not sufficient. However, for important subclasses of equations (17) it is possible to obtain characterizations of integrability of S(t). These classes contain those equations which admit an analytic resolvent, and parabolic equations with 2-regular kernels, as well as subordinated resolvents. To mention a result which is particularly striking, suppose A is the generator of a uniformly bounded cosine family Co(t), and let a(t) be of the form a(t) = a0 + a∞ t +

t 0

a1 (τ )dτ,

t > 0,

(41)

where a0 , a∞ ≥ 0, a1 ≥ 0 nonincreasing and log a1 (t) convex. This situation arises naturally in viscoelasticity; see Section 5.4. Then by the subordination principle the resolvent S(t) exists; it is integrable if and only if A is invertible and a(t) ≡ a∞ t. Observing that a(t) ≡ a∞ t corresponds to purely elastic materials, this result yields the remarkable fact that the presence of a whatsoever small viscoelasticity in the material stabilizes it strongly enough to obtain stability, i.e. integrability of the resolvent; see Section 10.3. A similar phenomenon can also be found in electrodynamics with memory; see Section 13.3. For problems of nonscalar type which are hyperbolic, integrability of S(t) is in general a too strong concept. However, for equations of variational type (31) we prove strong integrability, i.e. S(·)v ∗ ∈ L1 (⺢+ ; V ∗ ) for each v ∗ ∈ V ∗ ; cp. Section 10.4. This result applies to nonisotropic hyperbolic viscoelasticity. On the other hand, in the stable case resolvents for parabolic problems can even be expected to be uniformly integrable, i.e. S ∈ L1 (⺢+ ; B(X)) holds; Section 10.5 is devoted to this question. The general theory of (2) is presented in Sections 11 and 12. To be able to exploit the property of translation invariance which (2) enjoys, we study the solvability behaviour of (2) only in spaces of functions H(X) on the line which are translation invariant in the sense that the group of translations is strongly continuous and bounded in H(X). This leads to the concept of homogeneous spaces. For the convolution appearing in (2) to make sense, (2) is actually considered in differentiated form ∞ ∞ v(t) ˙ = A0 (τ )v(t ˙ − τ )dτ + dA1 (τ )v(t − τ ) + g(t), t ∈ ⺢, (42) 0

0

where A(t) = A0 (t) + A1 (t), A0 ∈ L1 (⺢+ ; B(Y, X)), A1 ∈ BV (⺢+ ; B(Y, X)). The concept for (2) corresponding to the ‘well-posedness’ of (1) is that of b-admissibility of a homogeneous space H(X), where b ∈ L1 (⺢) is a scalar function with nonvanishing Fourier transform ˜b(ρ) = 0 on ⺢, i.e. b ∈ W(⺢), the Wiener class. This means that for each function g = b ∗ h, with h ∈ H(X), there is a unique strong solution u ∈ H(X) of (42), and the solution operator G : g → u is bounded in H(X). Thus the scalar function b(t) measures the amount of time regularity of the forcing function g(t) needed for (42) to admit a strong solution. The main problem then consists in characterizing b-admissibility, and to obtain a representation of

Introduction

xxv

the solution operator G as a convolution, i.e. ∞ (Gg)(t) = G(τ )g(t − τ )dτ, −∞

t ∈ ⺢;

(43)

recall G(t) = S(t) if the resolvent S(t) is integrable. Formally the Fourier transform u ˜(ρ) of the solution u(t) of (42) is given by u ˜(ρ) = H(iρ)˜ g (ρ),

ρ ∈ ⺢,

(44)

where H(λ) again denotes the symbol of (1), (2), i.e. H(λ) =

1 −1 ˆ (I − A(λ)) . λ

(45)

Let the (real) spectrum of (42) be defined by 1 (iρ) ∈ B(Y, X) is not invertible}; Λ0 = {ρ ∈ ⺢ : iρ − iρAˆ0 (iρ) − dA

(46)

then it is natural to expect that (42) admits a solution if u ˜(ρ) = 0 on a neighborhood of Λ0 . This leads to localization in the frequency domain, where the spectrum σ(f ) of functions of subexponential growth becomes important; it is studied in some detail in Sections 0.5 and 0.6. By means of b-admissibility of the spaces HΛ (X) = {f ∈ H(X) : σ(f ) ⊂ Λ}, (47) where Λ ⊂ ⺢ is closed, and H(X) is homogeneous, the solvability behaviour of (42) can be adequately described. Necessary conditions for b-admissibility of CubΛ (X) are easily seen to be Λ ∩ Λ0 = ∅,

sup {|H(iρ)|B(X) + |˜b(ρ)H(iρ)|B(X,Y ) } < ∞.

(48)

ρ∈Λ

However, these are not always sufficient, but they are if Λ is compact (see Section 11.3), and for several important classes of kernels A(t) for arbitrary Λ. Section 12 is devoted entirely to the sufficiency of (48) for b-admissibility of HΛ (X) and the construction of Λ-kernels, i.e. kernels GΛ (t) representing the solution operator GΛ for HΛ (X) by means of (43). The class of equations (42) for which this works, includes parabolic ones, certain hyperbolic problems in Hilbert spaces, and subordinated equations of scalar type. b-admissibility of CubΛ (X) alone yields nice solvability behaviour of (42) w.r.t. periodic or almost periodic functions. For example, if Λ = (2π/ω)⺪ then for every ω-periodic function g there is a unique ω-periodic mild solution u of (42) and the Fourier coefficients un of u are determined by those of g according to un = H(2πin/ω)fn ,

n ∈ ⺪.

(49)

Similar relations are also valid for almost periodic or asymptotically almost periodic or weakly almost periodic functions on the line. If g ∈ CubΛ (X), 0 ∈ Λ and g(∞) = limt→∞ g(t) exists then u(t) also admits a limit as t → ∞ and u(∞) = H(0)g(∞).

(50)

xxvi

Introduction

There is also a satisfactory result on asymptotic equivalence of (42) and its version on ⺢+ , provided Λ0 = ∅, lim|λ|→∞ |H(λ)| < ∞ and the complex spectrum of (1), ˆ Σ0 = {λ ∈ ⺓+ : λ − λA(λ) ∈ B(Y, X) is not invertible},

(51)

is compact. In fact, if (42) admits a Λ-kernel, with Λ = ⺢, Cub (X) is b-admissible and the resolvent S(t) exists, then the solution of (42) is asymptotic to the solution of its local version on ⺢+ , whenever the latter is bounded. Finally, Section 13 contains several applications of the results of Chapter 3 to problems in viscoelasticity, heat conduction with memory and electrodynamics with memory, which have been introduced and discussed before in Sections 5 and 9. The text is concluded with a discussion and a bibliography of several subjects which are strongly related to the material presented in this book, but due to space considerations have not been treated in detail, here. This discussion includes the semigroup approach, and nonautonomous linear but also some nonlinear equations. Readers with special interest for nonlinear problems in viscoelasticity are refered to the recent monograph by Renardy, Hrusa and Nohel [289], which contains a detailed treatment of nonlinear one-dimensional models.

0

Preliminaries

This section contains some of the notations used throughout this book and collects some material scattered in the literature which is important for the theory to be developed but is also of independent interest. It covers the basic inversion theorems for the vector-valued Laplace transform. The Fourier-Carleman transform and the spectrum of vector-valued functions of subexponential growth are discussed in some detail. The study of Volterra equations in Banach spaces is begun with some elementary but nevertheless important existence results for equations with bounded kernels on finite intervals and is then continued with the Paley-Wiener theorems for the line and the halfline in the vector-valued case. 0.1 Some Notation Most notations used throughout this book are fairly standard in the modern mathematical literature. So ⺞, ⺪, ⺡, ⺢, ⺓ denote the sets of natural numbers, integers, rational numbers, real and complex numbers, respectively, and ⺞0 = ⺞ ∪ {0}, ⺢+ = [0, ∞), ⺓+ = {λ ∈ ⺓ : Re λ > 0}. If (M, d) is a metric space and N ⊂ M , ◦ ¯ , ∂N designate the interior, closure, boundary of N , respectively, and then N , N ¯r (x0 ) are the open d(x, N ) denotes the distance of x to N , while Br (x0 ) and B resp. closed balls with center x0 and radius r. X, Y, Z will always be Banach spaces with norms |·|X , |·|Y , |·|Z ; the subscripts will be dropped when there is no danger of confusion. B(X, Y ) denotes the space of all bounded linear operators from X to Y , B(X) = B(X, X) for short. The dual space of X is X ∗ = B(X, ⺛), where ⺛ = ⺢ or ⺛ = ⺓ is the underlying scalar field; < x, x∗ > designates the natural pairing between elements x ∈ X and x∗ ∈ X ∗ . If (xn ) ⊂ X converges to x ∈ X we write xn → x or limn→∞ xn = x, ∗ while xn x or w − limn→∞ xn = x mean weak convergence; similarly x∗n x∗ or w∗ − limn→∞ x∗n = x∗ stand for weak∗ -convergence of (x∗n ) ⊂ X ∗ to x∗ ∈ X ∗ . If A is a linear operator in X, D(A), R(A), N (A) denote domain, range, null space of A, respectively, while σ(A) and ρ(A) mean spectrum and resolvent set of A. σ(A) is further decomposed into σp (A), σc (A), σr (A), the point spectrum, continuous spectrum, and residual spectrum of A. The operator A∗ in X ∗ denotes the dual of A if it exists. If A is closed then D(A) equipped with the graph norm of A, |x|A = |x| + |Ax|, is a Banach space, for which the symbol XA is employed. For the concepts listed above as well as for general reference to operator theory we refer to Dunford and Schwarz [103], Hille and Phillips [180], and Kato [193]. Some frequently used function spaces are the following. If (M, d) is a metric space, and X a Banach space, then C(M ; X) denotes the space of all continuous functions f : M → X. Cb (M ; X) resp. Cub (M ; X) designate the spaces of all bounded continuous resp. bounded uniformly continuous functions f : M → X; these spaces become Banach spaces when normed by the sup-norm |f |0 = sup |f (t)|.

(0.1)

t∈M

1

2

Preliminaries

The space of all functions f : M → X which are uniformly Lipschitz-continuous is denoted by Lip(M ; X), and |f |Lip = sup |f (t) − f (s)|/d(t, s).

(0.2)

t=s

If (Ω, Σ, μ) is a measure space then Lp (Ω, Σ, μ; X), 1 ≤ p < ∞, denotes the space of all Bochner-measurable functions f : Ω → X such that |f (·)|p is integrable. This space is also a well-known Banach space when normed by |f |p = ( |f (t)|p dμ(t))1/p , (0.3) Ω

and functions equal a.e. are identified. Similarly, L∞ (Ω, Σ, μ; X) denotes the space of (equivalence classes of) Bochner-measurable essentially bounded functions f : Ω → X, and the norm is defined according to |f |∞ = ess sup |f (t)|.

(0.4)

t∈Ω

For Ω ⊂ ⺢n open, Σ the Lebesgue σ−algebra, μ the Lebesgue measure, we abbreviate to Lp (Ω; X). In this case W m,p (Ω; X) is the space of all functions f : Ω → X having distributional derivatives Dα f ∈ Lp (Ω; X) of order |α| ≤ m; the norm in W m,p (Ω; X) is |f |m,p = ( |Dα f |pp )1/p for 1 ≤ p < ∞, (0.5) |α|≤m

and |f |m,∞ = max |Dα f |∞ |α|≤m

for p = ∞,

¯ X) denotes the space of all functions f : Ω ¯ →X Also for Ω ⊂ ⺢n open, C m (Ω; α α which admit continuous partial derivatives ∂ f in Ω and ∂ f has continuous ¯ for each |α| ≤ m. The norm in its subspaces C m (Ω; ¯ X) and extension to Ω, b m ¯ Cub (Ω; X) is given by |f |m,0 = sup |∂ α f |0 . (0.6) |α|≤m

For f ∈ C(Ω; X) the support of f is defined by supp f = {x ∈ Ω : f (x) = 0}. A usual C0∞ (Ω; X) means the space of all functions f ∈ m≥1 C m (Ω; X) such that supp f ⊂ Ω is compact. Another space of interest is the space BV (J; X) where J is a closed interval. This space consists of all functions k : J → X of strongly bounded variation, i.e. N Var k|J = sup{ |k(tj ) − k(tj−1 )| : t0 < t1 < . . . < tN , tj ∈ J} j=1

(0.7)

Preliminaries

3

is finite. If J = [0, T ] or J = ⺢+ , elements of BV (J; X) are normalized throughout this book by the requirements k(0) = 0 and k(·) is left-continuous on J. Sometimes it is convenient to think of BV (J; X) as a subspace of BV (⺢; X) by extending functions k ∈ BV (J; X) by 0 to all of ⺢; the extension will be left-continuous again. By BV 0 (J; X), J = [0, T ] or J = ⺢+ , we denote the subspace of all functions k ∈ BV (J; X) which are continuous at 0, i.e. which satisfy 0 = k(0) = k(0+) = limt→0+ k(t). The subscript ‘loc’ assigned to any of the above function spaces means membership to the corresponding space when restricted to compact subsets of its domain. Usually if X = ⺛ is the underlying scalar field ⺛ = ⺓ or ⺛ = ⺢, the image space in the function space notation introduced above will be dropped. For example L1loc (⺢) denotes the space of all measurable scalar-valued functions which are integrable over each compact interval. Other function spaces will be introduced where they are needed for the first time; cp. the index. The Fourier transform of a function f ∈ L1 (⺢; X) is defined by ∞ ˜ e−iρt f (t)dt, ρ ∈ ⺢; (0.8) f (ρ) = −∞

it is well-known that f˜ : ⺢ → X is uniformly continuous and tends to 0 as |ρ| → ∞, by the Riemann-Lebesgue lemma. If X is a Hilbert space, by Parseval’s theorem the Fourier transform extends to a unitary operator on L2 (⺢; X). On the Schwartz space S(⺢; X) of all functions f ∈ C ∞ (⺢; X) such that each derivative of f decays faster than any polynomial, the Fourier transform is an isomorphism, and the inversion formula ˜ f˜(t) = 2πf (−t), t ∈ ⺢, (0.9) holds. The Laplace transform of a function f ∈ L1loc (⺢+ ; X) is denoted by f(λ) =

∞

e−λt f (t)dt,

Re λ > ω,

(0.10)

0

whenever the integral is absolutely convergent for Re λ > ω; see Section 0.2 for further discussions. The relation between the Laplace transform of f ∈ L1 (⺢; X), f (t) ≡ 0 for t < 0, and its Fourier transform is f˜(ρ) = fˆ(iρ),

ρ ∈ ⺢.

(0.11)

Observe that the symbol fˆ(λ) is also used for the Fourier-Carleman transform of f ∈ L1loc (⺢; X); see Section 0.5 for the definition of the latter. As usual we employ the star ∗ for the convolution of functions defined on the line but also on the halfline ∞ (f ∗ g)(t) = f (t − s)g(s)ds, t ∈ ⺢, (0.12) −∞

4

Preliminaries

e.g. for f, g ∈ L1 (⺢), and (f ∗ g)(t) =

t 0

f (t − s)g(s)ds,

t ∈ ⺢+ ,

(0.13)

e.g. for f, g ∈ L1 (⺢+ ). Observe that (0.12) and (0.13) are equivalent for functions which vanish for t < 0; therefore there will be no danger of confusion. For other symbols and notations the reader should consult the index. 0.2 Laplace Transform Let u ∈ BVloc (⺢+ ; X), where X denotes a complex Banach space. du is said to be exponential growth or Laplace transformable, if there is ω ∈ ⺢ such that ∞of−ωt e |du(t)| < ∞. In this case the Laplace transform of du, 0 ∞ du(λ) = e−λt du(t), Re λ ≥ ω, (0.14) 0

is well-defined, it is uniformly continuous and bounded in the closed right halfplane Re λ ≥ ω, and holomorphic in its interior. Similarly, if v ∈ L1loc (⺢+ ; X) is of ∞ −ωt exponential growth, i.e. 0 e |v(t)|dt < ∞ for some ω ∈ ⺢, we define ∞ v(λ) = e−λt v(t)dt, Re λ ≥ ω. (0.15) 0

Many of the properties of the scalar-valued Laplace transform are valid in the vector-valued case as well, some of them will be mentioned without poof here. The standard reference for the vector-valued Laplace transform is Chapter VI of Hille and Phillips [180]; for the classical Laplace transform we refer to Widder [339], [340] and Doetsch [100]. The function u(t) can be recovered from its Laplace transform via the complex inversion formula γ+iN 1 dλ 1 lim eλt du(λ) = (u(t) + u(t+)), t ≥ 0, (0.16) N →∞ 2πi γ−iN λ 2 where γ > ω is arbitrary; here the normalization of u(t) by u(0) = 0 and leftcontinuity is important. The limit in (0.16) is even uniform for t in any finite interval of continuity of u(t). Similarly v(t) can be recovered a.e. from vˆ(λ) by means of N 1 |ρ| (σ+iρ)t lim (1 − )e vˆ(σ + iρ)dρ = v(t), for a.a. t ≥ 0, (0.17) N →∞ 2π −N N and the limit in (0.17) exists also in L1 (J; X), for every finite interval J ⊂ ⺢+ ; here σ > ω is arbitrary. By means of Cauchy’s theorem, the complex inversion formula (0.16) leads to a simple characterization of functions which are holomorphic and bounded on each sector | arg z| ≤ θ, θ < θ0 .

Preliminaries

5

Theorem 0.1 Let f : (0, ∞) → X and θ0 ∈ (0, π/2]. Then the following are equivalent. (i) There is a function v(z) holomorphic for | arg z| < θ0 and bounded on each sector | arg z| ≤ θ < θ0 such that f (λ) = vˆ(λ) for each λ > 0; (ii) f (λ) admits holomorphic extension to the sector | arg λ| < π2 + θ0 , and λf (λ) is bounded on each sector | arg λ| ≤ π2 + θ, θ < θ0 . Proof: (⇒) Let v(z) be holomorphic on the sector | arg z| < θ0 ≤ π/2, bounded on each subsector | arg z| ≤ θ < θ0 , and define f (λ) = vˆ(λ), λ > 0. For |φ| < θ 0 consider the path γR = [0, R], R exp(i[0, φ]), [R, 0]eiφ ; by Cauchy’s theorem v(z)e−λz dz = 0, i.e. with R → ∞ we obtain γR

∞

vˆ(λ) =

v(t)e−λt dt =

0

∞

v(teiφ )e−λte eiφ dt, iφ

λ > 0.

0

This relation extends vˆ(λ) holomorphically to the sector | arg z| < θ0 + π/2, and gives the estimate |λˆ v(λ)| = |v(·eiφ )|∞ / cos(ρ + φ),

for all ρ = arg λ

satisfiying −π/2 − φ < ρ < π/2 − φ. Since φ ∈ (−θ0 , θ0 ) can be chosen arbitrarily, there follows boundedness of λˆ v (λ) on each sector | arg λ| ≤ θ + π/2, where θ < θ0 . (⇐) Suppose f (λ) admits holomorphic extension to | arg λ| < π/2 + θ0 , such that λf (λ) is bounded on each subsector | arg λ| ≤ φ < π/2 + θ0 . Consider the path ΓR consisting of the rays (∞, R]e−iφ , [R, ∞)eiφ connected by the part of the circle |λ| = R contained in | arg λ| ≤ φ, and define 1 v(z) = eλz f (λ)dλ, | arg z| < φ − π/2. 2πi ΓR Since this integral is absolutely convergent for | arg z| < φ − π/2, φ < π/2 + θ0 is arbitrary and v(z) is independent of φ and R > 0, by Cauchy’s theorem, there follows that v(z) is holomorphic on the sector | arg z| < θ0 . Choosing R = 1/|z|, a simple estimate shows the boundedness of v(z) on each subsector | arg z| ≤ θ < θ0 . Finally, for the Laplace transform of v(t) we obtain 1 vˆ(μ) = f (λ)(λ − μ)−1 dλ = f (μ), μ > R, 2πi ΓR by Cauchy’s theorem again. 2

Of interest here are also the real inversion formulae for the Laplace transform. The following one is due to Phillips; cp. Hille and Phillips [180], Chapter VI. lim e

σ→∞

−σt

∞ (−1)n (σ 2 t)n+1 (n) vˆ (σ) = v(t), n!(n + 1)! n=0

for a.a. t > 0,

(0.18)

6

Preliminaries

and the limit exists also in L1 (J; X), for every finite interval J ⊂ ⺢+ , it is uniform on any finite interval of continuity of v(t). Similarly, for u(t) we have lim e−σt

σ→∞

∞ (−1)n (σ 2 t)n (σt)2 (n) 1 du (σ) = (u(t) + u(t+)), n!(n + 2)! 2 n=0

t > 0,

(0.19)

where the limit is uniform on every finite interval of continuity of u(t). The real inversion formula (0.19) leads to the following extension of Widder’s characterizations of scalar-valued L∞ - and BV -functions in terms of their Laplace transform; see Widder [339]. Theorem 0.2 Let f : (0, ∞) → X. The following are equivalent. (i) There is u ∈ Lip(⺢+ ; X), u(0) = 0, such that f (λ) = du(λ) for all λ > 0; ∞ (ii) f ∈ C ((0, ∞); X) and sup{λn+1 |f (n) (λ)|/n! : λ > 0 , n ∈ ⺞0 } =: m∞ (f ) < ∞. If this is the case then |u|Lip = m∞ (f ). Theorem 0.3 Let f : (0, ∞) → X. The following are equivalent. (i) There is u ∈ BV (⺢+ ; X) such that f (λ) = du(λ) for all λ > 0; ∞ (ii) f ∈ C ((0, ∞); X) and sup

∞

λn |f (n) (λ)|/n! =: m1 (f ) < ∞.

λ>0 n=0

If this is the case then Var u|∞ 0 = m1 (f ). For the proof of Theorem 0.2 the following lemma is useful. Lemma 0.1 Suppose fn ∈ C 2 (J; X) satisfies fn → f in C(J; X) and |f¨n |0 ≤ M < ∞, for all n ∈ ⺞, where J = [0, a]. Then fn → f in C 1 (J; X), f˙ ∈ Lip(J; X), and |f˙|Lip ≤ M . Proof: Let g ∈ C 2 (J; X); by means of Taylor’s formula

h

g(t + h) = g(t) + hg(t) ˙ + 0

(h − s)¨ g (t + s)ds,

we obtain the estimate |h| |g| ˙ 0 ≤ 2|g|0 + (h2 /2)|¨ g |0 ,

h > 0,

which by the choice |h| = a/2(|g|0 /|g|2,0 )1/2 yields the interpolation inequality 1/2

1/2

|g| ˙ 0 ≤ c(a)|g|0 |g|2,0 ,

for all g ∈ Cb2 (J; X),

(0.20)

Preliminaries

7

with c(a) = (a/4+4/a). (0.20) applied to g = fn −fm implies fn → f in C 1 (J; X), and from |f˙n (t) − f˙n (s)| ≤ |f¨n |0 |t − s| ≤ M (t − s), t, s ∈ J, with n → ∞ the result follows. 2

For the proof of Theorem 0.3 we take advantage of Lemma 0.2 Let J = [0, a]; suppose fn ∈ C 2 (J; X) satisfies fn → f in C(J; X) and |f¨n |1 ≤ M < ∞. Then f (t) is absolutely continuous on J, f˙ ∈ BV (J; X), f˙n (t) → f˙(t) for a.a. t ∈ J, f˙n → f˙ in L1 (J; X), and Var f˙n |J ≤ M . t Proof: Let kn (t) = 0 |f¨n (s)|ds, t ∈ J, n ∈ ⺞; these functions are nondecreasing and uniformly bounded by M . Helly’s selection theorem therefore provides a subsequence (nk ) such that knk (t) → k(t) pointwise; the limit function k(t) is nondecreasing and bounded by M as well. Let t ∈ J be a point of continuity of k(t), and let Dh+ denote the right difference operator Dh+ g(t) = (g(t + h) − g(t))/h; then |f˙n (t) − f˙m (t)| ≤ |f˙n (t) − Dh+ fn (t)| + |Dh+ (fn (t) − fm (t))| + |f˙m (t) − D+ fm (t)| h

≤ [kn (t + h) − kn (t)] + 2h−1 |fn − fm |0 + [km (t + h) − km (t)]. Given ε > 0 choose first h > 0 small enough such that k(t + h) − k(t) ≤ ε/8; fix h > 0 and choose n, m so large that |fn − fm |0 ≤ εh/8, and finally enlarge n = nk , m = nl such that |kn (s) − k(s)| ≤ ε/8, and |km (s) − k(s)| ≤ ε/8, for s = t, s = t + h. This gives |f˙n (t) − f˙m (t)| ≤ 2(k(t + h) − k(t)) + 4 · ε/8 + 2h−1 εh/8 ≤ ε, i.e. f˙nk (t) → g(t) for each continuity point of k(t). Since for two such points |f˙n (t) − f˙n (s)| ≤ kn (t) − kn (s), we obtain in the limit |g(t) − g(s)| ≤ k(t) − k(s), i.e. g ∈ BV (J; X). Finally, by Lebesgue’s theorem

t

fnk (t) = fnk (t0 ) + t0

f˙nk (s)ds → f (t0 ) +

t

g(s)ds = f (t), t0

hence f (t) is absolutely continuous on J, and g(t) = f˙(t) a.e. Repeating these arguments with an arbitrary subsequence we may conclude f˙n → f˙ a.e. and in L1 (J; X). 2

8

Preliminaries

The arguments in the proofs of Theorems 0.2 and 0.3 are quite similar, therefore they are carried through simultaneously. Proofs of Theorems 0.2 and 0.3: (i) ⇒ (ii) of Theorem 0.2. Let u ∈ Lip(⺢+ ; X), u(0) = 0, and define f (λ) = λˆ u(λ) = du(λ); then ∞ f (n) (λ) = (−t)n e−λt du(t), λ > 0, 0

hence |f

(n)

∞

(λ)| =

n −λt

t e 0

|du(t)| ≤ |u|Lip

∞ 0

tn e−λt dt = |u|Lip n!λ−(n+1) ,

i.e. m∞ (f ) ≤ |u|Lip < ∞. (i) ⇒ (ii) of Theorem 0.3. If u ∈ BV (⺢+ ; X), u(0) = 0, we obtain similarly ∞ λn (n) |f (λ)| n! n=0

∞ ∞ ∞ λn ∞ n −λt (tλ)n −λt ≤ t e |du(t)| = ( )e |du(t)| n! 0 n! 0 n=0 n=0 ∞ = |du(t)| = Var u|∞ 0 , 0

u|∞ 0

hence m1 (f ) ≤ Var < ∞. (ii) ⇒ (i) in both theorems. Observe first that f (λ) admits holomorphic extension to the open right halfplane ⺓+ , as the series expansion f (z) =

∞

(z − λ)n f (n) (λ)/n!,

Re z > 0, |z − λ| < λ,

n=0

and the estimates |f (z)| ≤

∞

|z − λ|n |f (n) (λ)|/n! ≤ m1 (f ),

n=0

resp. |f (z)| ≤ m∞ (f )

∞

|(z − λ)|n λ−(n+1) =

n=0

m∞ (f ) m∞ (f ) →λ→∞ , λ − |z − λ| Re z

show. Define functions vn by means of vn (t) = e

−nt

∞ (n2 t)k+1 (−1)k (k) f (n), (k + 1)!k!

t ≥ 0;

k=0

then

∞ (nt)k+1 |vn (t)| ≤ e−nt ( )m∞ (f ) ≤ m∞ (f ), (k + 1)! k=0

Preliminaries

9

i.e. |vn |0 ≤ m∞ (f ), and ∞ |vn |1 ≤ ( k=0

∞

e−nt tk+1 dt)

0

n2k+2 |f (k) (n)| ≤ m1 (f ). (k + 1)!k!

The Laplace transform of vn (t) is easily computed to the result vˆn (λ) = (

n 2 nλ ) f( ), n+λ n+λ

Re λ > 0,

hence vˆn (λ) → f (λ) as n → ∞, uniformly on compact subsets of ⺓+ . Define wn (t) by t

wn (t) =

0

(t − s)vn (s)ds,

t ≥ 0, n ∈ ⺞;

then w ˆn (λ) = vˆn (λ)/λ2 , Re λ > 0, hence by boundedness of f (λ) on halfplanes Re λ ≥ σ > 0, the complex inversion formula for the Laplace transform yields σ+i∞ wn (t) = (2πi)−1 vˆn (λ)eλt dλ/λ2 , t ≥ 0. σ−i∞

Next Lebesgue’s theorem implies wn (t) →n→∞ w(t) = (2πi)−1

σ+i∞

f (λ)eλt dλ/λ2 ,

t ≥ 0,

σ−i∞

uniformly on bounded intervals J = [0, a] ⊂ ⺢+ , where the boundedness of f (λ) on Re λ ≥ σ > 0 was used again. We are in position now to apply Lemma 0.1 resp. Lemma 0.2. In fact, if 1 m∞ (f ) < ∞, Lemma 0.1 yields wn → w in Cloc (⺢+ ; X), and w˙ ∈ Lip(⺢+ ; X), as well as |w| ˙ Lip ≤ m∞ (f ); in case m1 (f ) < ∞, Lemma 0.2 implies wn → w in 1,1 Wloc (⺢+ ; X) and w˙ ∈ BV (⺢+ ; X), Var w| ˙ ∞ ˙ and 0 ≤ m1 (f ). Setting u(t) = w(t), observing that w(0) = w(0) ˙ = 0, the assertions of Theorems 0.2 and 0.3 follow. 2

In case X has the Radon-Nikodym property, then a Lipschitz function has an L∞ -derivative a.e. In this case Theorem 0.2 becomes Corollary 0.1 Suppose X has the Radon-Nikodym property, and let f : (0, ∞) → X. Then the following are equivalent. (i) There is v ∈ L∞ (⺢+ ; X) such that f (λ) = v(λ) for all λ > 0; (ii) f ∈ C ∞ ((0, ∞); X) and m∞ (f ) < ∞. If this is the case then |v|∞ = m∞ (f ). Theorem 0.2 as well as Corollary 0.1 are due to Arendt [10]. In practice m1 (f ) as well as m∞ (f ) are difficult to estimate, and so Theorems 0.2 and 0.3 are not easy to apply. However, quite often one encounters functions

10

Preliminaries

g(λ), holomorphic in the right halfplane Re λ > 0 and such that λg(λ) is bounded there. In general it is not true that this already implies m∞ (f ) < ∞ as one might expect; see Desch and Pr¨ uss [91]. However, the following result holds. Propositon 0.1 Suppose g : ⺓+ → X is holomorphic and satisfies |λg(λ)| + |λ2 g (λ)| ≤ M

for all Re λ > 0,

(0.21)

Then m∞ (g) < ∞. Proof: To estimate g (n) (λ) for n > 1, we use the Cauchy integral representation g (λ) =

1 2πi

ε+i∞

g (z)(λ − z)−1 dz,

λ > ε.

ε−i∞

Differentiating this formula (n − 1)-times yields g (n) (λ) = (−1)(n−1) (n − 1)!(2πi)−1

ε+i∞

g (z)(λ − z)−n dz,

λ > ε.

ε−i∞

By means of (0.21) we obtain |g (n) (λ)| ≤

∞

≤

|ε + iρ|−2 |λ − ε − iρ|−n dρ −∞ ∞ (n − 1)!(M/2π)(λ − ε)−n (ε2 + ρ2 )−1 dρ

≤

(n − 1)!(M/2π)(λ − ε)−n (π/ε).

(n − 1)!(M/2π)

−∞

Choosing ε = λ/n the latter gives |g (n) (λ)| ≤ n!(M/2)λ−(n+1) (1 − 1/n)−n ≤ n!(M e/2)λ−(n+1) , thereby proving m∞ (g) < ∞. 2

If only λg(λ) is bounded we can prove the following weaker result which nevertheless is useful as well. Proposition 0.2 Suppose g : ⺓+ → X is holomorphic and satisfies |λg(λ)| ≤ M,

Re λ > 0.

(0.22)

Then there is u ∈ C(⺢+ ; X) with u(0) = 0 such that g(λ) = λˆ u(λ), Re λ > 0; moreover, there is a constant L > 0 such that |u(t) − u(s)| ≤ L(t − s)[1 + log(t/(t − s))]

for all t > s ≥ 0.

(0.23)

Preliminaries

11

Proof: We define −1

t ≥ 0,

g(λ)eλt dλ/λ,

u(t) = (2πi)

Γε,r

(0.24)

where Γε,r denotes the contour ε + i(−∞, −r), ε + rei[−π/2,π/2] , ε + i(r, ∞), with ε, r > 0. Observe that the integral is absolutely convergent, by virtue of (0.22). Clearly, the definition of u(t) is independent of ε, r > 0, and contracting the contour in the right halfplane it follows that u(0) = 0, by Cauchy’s theorem. Fix t > s ≥ 0 and estimate by means of (0.22) to the result |u(t) − u(s)| ≤ (M/2π) |eλt − eλs ||dλ|/|λ|2 . Γε,r

Letting ε → 0 this yields |u(t) − u(s)|

≤ (M/π)[

∞

r

|eiρt − eiρs |dρ/ρ2 +

∞

π/2

0

iφ

iφ

|erte − erse |dφ/r]

| sin(ρ(t − s)/2)|dρ/ρ2 + ert (t − s)π/2] r ∞ ≤ M (t − s)[π −1 | sin τ |dτ /τ 2 + ert /2]. ≤ (M/π)[2

r(t−s)/2

Choosing r = 2/t we obtain |u(t) − u(s)|

≤ M (t − s)[π

−1

∞ 1−s/t

| sin τ |dτ /τ 2 + e2 /2]

≤ L(t − s)[1 + log(t/(t − s))], what was to be proved. Finally, we have by means of Cauchy’s theorem and Fubini’s theorem ∞ −λt −1 λˆ u(λ) = λ e u(t)dt = (2πi) g(μ)λ(λ − μ)−1 dμ/μ 0

= =

(2πi)−1 (2πi)−1

Γε,r

Γε,r

Γε,r

=

g(λ)

g(μ)(1 + μ/(λ − μ))dμ/μ g(μ)(λ − μ)−1 dμ

for all Re λ > r + ε. 2

Combining Propositions 0.1 and 0.2 with Theorem 0.2 we obtain the following result on inversion of the vector-valued Laplace transform. Theorem 0.4 Suppose g : ⺓+ → X is holomorphic and satisfies |λn+1 g (n) (λ)| ≤ M,

for Re λ > 0, 0 ≤ n ≤ k + 1,

(0.25)

12

Preliminaries

where k ≥ 0. Then there is a function u ∈ C k ((0, ∞); X) such that g(λ) = u ˆ(λ) for Re λ > 0. Moreover, there is a constant C > 0 such that |tn u(n) (t)| ≤ C,

for t > 0, 0 ≤ n ≤ k,

(0.26)

and |tk+1 u(k) (t) − sk+1 u(k) (s)| ≤ C|t − s|[1 + log(t/(t − s))],

0 ≤ s < t < ∞. (0.27)

Proof: For n ≤ k + 1 we define gn (λ) = λn g (n) (λ); then for n ≤ k, gn (λ) satisfies the assumptions of Proposition 0.1, hence there are functions un ∈ C(⺢+ ; X) with un (0) = 0 such that gn (λ) = λˆ un (λ) and there is L > 0 such that |un (t) − un (s)| ≤ L|t − s|,

for all t, s ∈ ⺢+ , 0 ≤ n ≤ k.

Proposition 0.2 yields uk+1 ∈ C(⺢+ ; X) with uk+1 (0) = 0 such that gk+1 (λ) = λˆ uk+1 (λ) and |uk+1 (t) − uk+1 (s)| ≤ L(t − s)[1 + log(t/(t − s))], Since

0 ≤ s < t < ∞.

gn (λ) = nλn−1 g (n) (λ) + λn g (n+1) (λ) = (ngn (λ) + gn+1 (λ))/λ

for all Re λ > 0 and 0 ≤ n ≤ k, we obtain u ˆ n (λ) = ((n − 1)ˆ un (λ) + u ˆn+1 (λ))/λ, which implies −tun (t) = (n − 1)

t 0

un (τ )dτ +

t 0

un+1 (τ )dτ,

t > 0, 0 ≤ n ≤ k,

by uniqueness of the Laplace transform. This identity shows un ∈ C 1 ((0, ∞); X) and −tu n (t) = nun (t) + un+1 (t), t > 0, 0 ≤ n ≤ k. Let u(t) = u 0 (t); then |u(t)| = |u1 (t)|/t ≤ L, hence u ˆ(λ) = λˆ u0 (λ) = g0 (λ) = g(λ),

Re λ > 0,

and un (t) = (−1)n (tn u(t))(n−1) as is easily seen by induction. The assertion now follows from the properties of un (t). 2

Propositions 0.1, 0.2 and Theorem 0.4 are taken from Pr¨ uss [278]. 0.3 Volterra Equations with Bounded Kernels Let X be a Banach space and consider the linear Volterra equation t u(t) = f (t) + dK(τ )u(t − τ ), t ∈ J, 0

(0.28)

Preliminaries

13

where K ∈ BV 0 (J; B(X)), f ∈ C(J; X), and J = [0, T ]. Along with (0.28) we also consider the resolvent equations for (0.28) in B(X)

t

L(t) = K(t) + 0

and

t

L(t) = K(t) + 0

dK(τ )L(t − τ ),

t ∈ J,

(0.29)

L(t − τ )dK(τ ),

t ∈ J.

(0.30)

The following result is basic. Theorem 0.5 Suppose K ∈ BV 0 (J; B(X)). Then there is a unique function L ∈ BV 0 (J; B(X)) such that (0.29) and (0.30) are satisfied. L(t) can be obtained via the Neumann series L(t) =

∞

Kn (t),

t ∈ J,

(0.31)

n=0

where K0 (t) = K(t), Kn+1 (t) =

t 0

dK(τ )Kn (t − τ ),

t ∈ J, n ∈ ⺞0 .

(0.32)

Proof: (Uniqueness) Let L1 ,L2 ∈ L1 (J; B(X)) be solutions of (0.29), (0.30), respectively. Then (L2 − K) ∗ L1 = (L2 ∗ dK) ∗ L1 = L2 ∗ (dK ∗ L1 ) = L2 ∗ (L1 − K), hence K ∗ L1 = L2 ∗ K, and consequently 1 ∗ L1 = 1 ∗ L2 . This implies L1 = L2 by differentiation, and so the solution L of (0.29), (0.30) is unique if it exists. (Existence) Let k(t) =Var K|t0 ; then k ∈ BV 0 (J) and by induction we obtain that Kn defined by (0.32) belongs to BV 0 (J; B(X)), and Var Kn |ts ≤ kn (t)−kn (s), where k0 (t) = k(t), and kn+1 (t) = (dk ∗ kn )(t). Let ω ≥ 0 be so large that T η = 0 e−ωt dk(t) < 1; such ω certainly exists since k(0) = k(0+) = 0. Then by T induction 0 e−ωt dkn (t) ≤ η n+1 , n ∈ ⺞0 , and therefore Var Kn |T0 ≤ kn (t) ≤ eωT

T 0

e−ωt dkn (t) ≤ η n+1 eωT ,

n ∈ ⺞0 .

This shows that the series defined by (0.31) converges absolutely in the norm of BV 0 (J; B(X)), and so L ∈ BV 0 (J; B(X)) exists. Since L − K = L − K0 =

∞ n=1

Kn = dK ∗

∞ n=1

Kn−1 = dK ∗ L

14

Preliminaries

and dK ∗ Kn = Kn ∗ dK for all n ∈ ⺞0 , L(t) satisfies the resolvent equations (0.29) and (0.30). The proof is complete. 2

The kernel L(t) is called the resolvent kernel associated with K(t). If u(t) is a continuous solution of (0.28), convolving (0.28) with dL(t) and adding the result to (0.28) we obtain the variation of parameters formula

t

u(t) = f (t) + 0

dL(τ )f (t − τ ),

t ∈ J.

(0.33)

Thus as a consequence of Theorem 0.5 we have Corollary 0.2 Suppose K ∈ BV 0 (J; B(X)) and let L ∈ BV 0 (J; B(X)) denote the resolvent kernel associated with K. Then for any f ∈ C(J; X) with f (0) = 0, there is a unique solution u ∈ C(J; X) of (0.28) which is given by the variation of parameters formula (0.33). Moreover, the solution map f → u preserves the spaces Lp (J; X), 1 ≤ p ≤ ∞, and BV 0 (J; X) and the estimates |u|p ≤ (1 + l(T ))|f |p , Var u|T0

≤ (1 +

f ∈ Lp (J; X),

l(T ))Var f |T0 ,

1 ≤ p ≤ ∞, 0

f ∈ BV (J; X),

(0.34) (0.35)

hold, where l(t) denotes the resolvent kernel of k(t), i.e. the solution of the scalar equation l = k + dk ∗ l. Observe that the solution of (0.28) with f (t) ≡ x is x + L(t)x, which is in general not continuous, but left-continuous only. (0.34) and (0.35) follow directly from (0.33), observing that l(T ) ≥Var L|T0 . The resolvent kernel L(t) associated with K(t) is also useful in connection with the operator-valued Volterra equations

t

U (t) = F (t) + 0

and

t

V (t) = G(t) + 0

dK(τ )U (t − τ ),

t ∈ J,

(0.36)

V (t − τ )dK(τ ),

t ∈ J.

(0.37)

dL(τ )F (t − τ ),

t ∈ J,

(0.38)

G(t − τ )dL(τ ),

t ∈ J.

(0.39)

In fact, the solution of (0.36) is given by

t

U (t) = F (t) + 0

while that of (0.37) is obtained via V (t) = G(t) + 0

t

Preliminaries

15

Observe that by application to a vector x ∈ X, (0.36) reduces to (0.28) and (0.38) to (0.33); however this is not the case for (0.37). This leads to a slightly different behaviour of (0.36) and (0.37). Corollary 0.3 Let Z denote another Banach space, K ∈ BV 0 (J; B(X)) and let L ∈ BV 0 (J; B(X)) be the resolvent kernel associated with K. Then (i) for every F ∈ Cs (J; B(Z; X)), F ∈ C(J; B(Z; X)), F ∈ BV (J; B(Z; X)), with F (0) = 0, there is a unique solution U ∈ Cs (J; B(Z; X)), U ∈ C(J; B(Z; X)), U ∈ BV 0 (J; B(Z; X)), of (0.36), respectively, and it is given by (0.38); (ii) for every G ∈ C(J; B(X; Z)) or G ∈ BV (J; B(X; Z)), with G(0) = 0, there is a unique solution V ∈ C(J; B(X; Z)) or V ∈ BV 0 (J; B(X; Z)), of (0.37), respectively, and it is given by (0.39). Here, by Cs (J; B(Z; X)) we mean the space of all strongly continuous families {F (t)}t∈J ⊂ B(Z; X), i.e. the functions F (·)z are continuous in X for each z ∈ Z. 0.4 Convolution Equations with Bounded Kernels on the Line Let X denote again a Banach space, and K ∈ L1 (⺢; B(X)). In this subsection, we consider the convolution equation on the line ∞ u(t) = f (t) + K(τ )u(t − τ )dτ, t ∈ ⺢, (0.40) −∞

where f ∈ Lp (⺢; X), 1 ≤ p ≤ ∞. Suppose (0.40) admits a unique solution u ∈ Cub (⺢; X) for each given f ∈ Cub (⺢; X). Then by the translation invariance of (0.40), the solution of (0.40) for any τ -periodic continuous f must be τ -periodic. In particular, it must be constant whenever f is constant. Now choose f (t) = eiρt y, ρ ∈ ⺢, y ∈ X; then v(t) = u(t)e−iρt satisfies v = y + Kρ ∗ v, where Kρ (t) = e−iρt K(t); hence v(t) ≡ x ˜ This and so u(t) ≡ eiρt x, for some x ∈ X; (0.40) then implies x = y + K(ρ)x. ˜ shows that I − K(ρ) is invertible for every ρ ∈ ⺢. We state this necessary condition as Proposition 0.3 Let K ∈ L1 (⺢; B(X)), and suppose (0.40) admits a unique ˜ is invertible for solution u ∈ Cub (⺢; X), for every f ∈ Cub (⺢; X). Then I − K(ρ) each ρ ∈ ⺢. Next consider the whole-line resolvent equations ∞ R(t) = K(t) + K(τ )R(t − τ ), −∞

and

∞

R(t) = K(t) + −∞

R(τ )K(t − τ ),

t ∈ ⺢,

(0.41)

t ∈ ⺢.

(0.42)

16

Preliminaries

Suppose (0.41) and (0.42) admit a solution R ∈ L1 (⺢; B(X)). Then R is necessarily unique and the unique solution u of (0.40) is given by the variation of parameters formula ∞ u(t) = f (t) + R(τ )f (t − τ )dτ, t ∈ ⺢. (0.43) −∞

It is clear that (0.43) leaves invariant Lp (⺢; X), 1 ≤ p ≤ ∞, as well as Cub (⺢; X). It turns out that the necessary condition obtained in Proposition 0.3 is also sufficient for existence of R ∈ L1 (⺢; B(X)). This is the main result of this subsection; in the case X = ⺓ it is due to Paley and Wiener [266], while the vector-valued case was proved in Pr¨ uss [271] and Gripenberg [154]; for more general versions see Gripenberg [155], and Gripenberg, Londen and Staffans [156]. Theorem 0.6 Let K ∈ L1 (⺢; B(X)). Then (0.41), (0.42) admit a unique solution ˜ R ∈ L1 (⺢; B(X)) if and only if I − K(ρ) is invertible for each ρ ∈ ⺢. Proof: (⇒) Let R ∈ L1 (⺢; B(X)) be a solution of (0.41), (0.42); then Fourier transform yields ˜ ˜ ˜ ˜ (I + R(ρ))(I − K(ρ)) = (I − K(ρ))(I + R(ρ)) = I,

ρ ∈ ⺢,

˜ and so I − K(ρ) is invertible for each ρ ∈ ⺢. (⇐) Let F ∈ Cub (⺢; B(X)) be defined by −1 ˜ ˜ F (ρ) = K(ρ)(I − K(ρ)) ,

ρ ∈ ⺢.

(0.44)

˜ We are going to show that F (ρ) = R(ρ), ρ ∈ ⺢, for some R ∈ L1 (⺢; B(X)); once this is known to be true it then follows easily that R solves (0.41), (0.42), by uniqueness of the Fourier transform. (i) Let ϕ ∈ L1 (⺢) be defined by ϕ(t) = 2(πt2 )−1 sin(3t/2) sin(t/2), t ∈ ⺢; a simple computation yields ⎧ for |ρ| ≤ 1 ⎨ 1 2 − |ρ| for 1 ≤ |ρ| ≤ 2 ϕ(ρ) ˜ = ⎩ 0 for |ρ| ≥ 2. Define ϕε (t) = εϕ(εt), t ∈ ⺢; then ϕ˜ε (ρ) = ϕ(ρ/ε), ˜ ρ ∈ ⺢, and with ρn = 3nε, |n| ≤ N , we obtain the partition of unity N

ϕ˜ε (ρ − ρn ) + ψN (ρ) = 1,

ρ ∈ ⺢,

−N

where

⎧ ⎨ 0 ψN (ρ) = |ρ|/ε − 3N − 1 ⎩ 1

for |ρ| ≤ (3N + 1)ε, for (3N + 1)ε ≤ |ρ| ≤ (3N + 2)ε, for |ρ| ≥ (3N + 2)ε.

(0.45)

Preliminaries

17

By means of this partition of unity we obtain the decomposition F (ρ) =

N

ϕ˜ε (ρ − ρn )F (ρ) + ψN (ρ)F (ρ),

ρ ∈ ⺢,

−N

and it is sufficient to show that each term of the right hand side is the Fouriertransform of an L1 (⺢; B(X))-function. (ii) Define Kn (t) by ˜ n )](I − K(ρ ˜ n ))−1 , Kn (t) = [((ϕ2ε eiρn · ) ∗ K)(t) − ϕ2ε (t)eiρn t K(ρ

t ∈ ⺢;

then

|Kn |1

∞

∞

K(τ )(ϕ2ε (t − τ ) − ϕ2ε (t))eiρn (t−τ ) dτ |dt ∞ ≤ M |K(τ )|( |ϕ2ε (t − τ ) − ϕ2ε (t)|dt)dτ −∞ −∞ ∞ ∞ = M |K(τ )|( |ϕ(t − 2ετ ) − ϕ(t)|dt)dτ −∞ −∞ ≤ 2M |ϕ|1 ( |K(τ )|dτ ) + M |K|1 · sup√ |ϕ(· − h) − ϕ|1 , √ ≤ M

−∞ ∞

|

−∞

|τ |≥1/ ε

|h|≤2 ε

−1 ˜ ˜ where M = sup{|(I − K(ρ)) | : ρ ∈ ⺢} < ∞ by assumption and since |K(ρ)| →0 as |ρ| → ∞, by the Riemann-Lebesgue lemma. Choosing ε > 0 small enough we obtain |Kn |1 ≤ 1/2 for each |n| ≤ N . The Fourier transform of Kn (t) is easily evaluated to the result

˜ n (ρ) = ϕ˜2ε (ρ − ρn )(K(ρ) ˜ ˜ n ))(I − K(ρ ˜ n ))−1 , K − K(ρ

ρ ∈ ⺢,

hence −1 ˜ ˜ ˜ ˜ n ))−1 (I−K ˜ n (ρ))−1 , Fn (ρ) = ϕ˜ε (ρ−ρn )K(ρ)(I− K(ρ)) = ϕ˜ε (ρ−ρn )K(ρ)(I− K(ρ

with ϕ˜2ε (ρ) ≡ 1 on supp ϕ˜ε . Since |Kn |1 ≤ 1/2, Fn (ρ) is the Fourier transform of Rn ∈ L1 (⺢; B(X)) given by ˜ n ))−1 Rn = ϕε eiρn ∗ K ∗ (I − K(ρ

∞

Kn∗l .

l=0

(iii) Let χ(t) = 2(πt2 )−1 sin2 (t/2) denote Fejer’s kernel, define dilations of them by χN (t) =3N εχ(3N εt), and let KN = χN ∗ K. Then KN → K in L1 (⺢; B(X)) as N → ∞, ε > 0 fixed, and

˜ (1 − |ρ|/3N ε)K(ρ) for |ρ| ≤ 3N ε, ˜ N (ρ) = K 0 otherwise.

18

Preliminaries

˜ N (ρ) = 0 on supp ψN , and therefore In particular K −1 ˜ ˜ FN (ρ) = ψN (ρ)K(ρ)(I − K(ρ))

˜ = (K(ρ) −

N

˜ ˜ ˜ N (ρ)))−1 ϕ˜ε (ρ − ρn )K(ρ))(I − (K(ρ) −K

−N

is the Fourier transform of a function RN ∈ L1 (⺢; B(X)) as at the end of step (ii), provided N is chosen so large that |K − KN |1 < 1 holds. The proof is now complete. 2

As indicated in front of Theorem 0.6, we obtain the following result on solvability properties of (0.40) as a consequence of this result. ˜ Corollary 0.4 Let K ∈ L1 (⺢; B(X)) be such that I − K(ρ) is invertible for ρ ∈ ⺢. Then for every f ∈ Lp (⺢; X) there is a unique solution u ∈ Lp (⺢; X) of (0.40), where 1 ≤ p ≤ ∞. If R ∈ L1 (⺢; B(X)) denotes the whole-line resolvent of (0.40) the solution u of (0.40) is given by (0.43). Moreover, the spaces Cub (⺢; X) and Cb (⺢; X) are also invariant under the solution map f → u. In the Volterra case we have supp K ⊂ ⺢+ ; this does not necessarily imply supp R ⊂ ⺢+ , in case R ∈ L1 (⺢; B(X)) exists. In fact, R ∈ L1 (⺢+ ; B(X)) yields ˆ ˆ ˆ ˆ (I − K(λ))(I − R(λ)) = (I − R(λ))(I − K(λ) = I,

Re λ ≥ 0,

ˆ i.e. I − K(λ) must necessarily be invertible for all Re λ ≥ 0, not only on the imaginary axis. It turns out that this condition is also sufficient; this is the PaleyWiener lemma on the halfline. The argument used in the proof is a straightforward adaption of the original idea of Paley and Wiener [266]; see also Gripenberg, Londen, and Staffans [156]. Theorem 0.7 Let K ∈ L1 (⺢+ ; B(X)), and extend K(t) to all of ⺢ by 0. Then (0.41), (0.42) admit a unique solution R ∈ L1 (⺢+ ; B(X)) (R(t) ≡ 0 for t < 0) if ˆ and only if I − K(λ) is invertible for each Re λ ≥ 0. ˆ ˜ Proof: Extend K(t) by 0 to all of ⺢. Since I − K(iρ) = I − K(ρ) is invertible for all ρ ∈ ⺢, by Theorem 0.6 there is a solution R ∈ L1 (⺢; B(X)) of (0.41), (0.42). Let R+ (t) = R(t)e0 (t), R− (t) = R(−t) − R+ (−t), t ∈ ⺢, where e0 (t) denotes the Heaviside function. Then R+ and R− vanish for t < 0, hence their Laplace transforms are well-defined, analytic for Re λ > 0, bounded and continuous on ¯ + , and tend to zero as |λ| → ∞. Define ⺓

−1 ˆ ˆ ˆ + (λ), Re λ > 0 K(λ)(I − K(λ)) −R G(λ) = ˆ R− (−λ), Re λ < 0; ˜ then G(iρ + 0) = R(ρ) − R˜+ (ρ) = R˜− (ρ) = G(iρ − 0), ρ ∈ ⺢, i.e. G(λ) is entire by Morera’s theorem, (cp. eg. Conway [59]), bounded and G(λ) → 0 as |λ| → ∞.

Preliminaries

19

−1 ˆ ˆ Liouville’s theorem yields G(λ) ≡ 0, hence Rˆ+ (λ) = K(λ)(I − K(λ)) , Re λ ≥ 0, i.e. R = R+ by uniqueness of the Fourier transform. 2

0.5 The Spectrum of Functions of Subexponential Growth Let f ∈ L1loc (⺢; X) be of subexponential growth, where X denotes a complex Banach space; by this we mean ∞ e−ε|t| |f (t)|dt < ∞, for each ε > 0. −∞

The Fourier-Carleman transform fˆ of f is defined by ∞ e−λt f (t)dt, Re λ > 0 0 fˆ(λ) = 0 − −∞ e−λt f (t)dt, Re λ < 0. Obviously, fˆ(λ) is holomorphic in ⺓ \ i ⺢. Define ρ(f ) = {ρ ∈ ⺢ : fˆ(λ) is analytically extendable to some Bε (iρ)};

(0.46)

then σ(f ) = ⺢ \ ρ(f ) is called the spectrum of f . The Fourier-Carleman transform and this approach of defining the spectrum of functions in the scalar case are due to Carleman [37]; see also Katznelson [194]. Example 0.1 (i) Suppose f ∈ L1loc (⺢; X) is τ -periodic. Then an easy computation yields τ fˆ(λ) = (1 − e−λτ )−1 e−λt f (t)dt, Re λ = 0, (0.47) 0

i.e. fˆ(λ) extends to a meromorphic function with simple poles at most at λn = 2πin/τ , n ∈ ⺪. The residuum at λ = λn is given by 1 τ −2πint/τ ˆ Resf (λ)|λ=λn = fn = e f (t)dt, n ∈ ⺪, τ 0 the nth Fourier coefficient of f . Therefore we have σ(f ) = {2πn/τ : n ∈ ⺪, fn = 0} ⊂ (2π/τ )⺪.

(0.48)

(ii) Let f ∈ L1 (⺢; X) and let f˜(ρ) denote its Fourier transform ∞ f˜(ρ) = f (t)e−iρt dt, ρ ∈ ⺢.

(0.49)

−∞

Then lim (fˆ(iρ + σ) − fˆ(iρ − σ)) = lim

σ→0+

σ→0+

∞ −∞

f (t)e−iρt e−σ|t| dt = f˜(ρ),

20

Preliminaries

for every ρ ∈ ⺢; therefore ρ0 ∈ σ(f ) implies f˜(ρ) ≡ 0 in a neighborhood of ρ0 , and so ρ0 ∈ supp f˜. Conversely, if ρ0 ∈ supp f˜ then fˆ(λ) is continuous on a ball Bε (iρ0 ), hence by Morera’s theorem (cp. Conway [59]), fˆ(λ) is holomorphic there, i.e. ρ0 ∈ σ(f ). This implies the relation σ(f ) = supp f˜, for all f ∈ L1 (⺢; X). (0.50) 2

Below we will show that (0.50) holds for all f ∈ L1loc (⺢; X) of polynomial growth provided the Fourier transform of f is understood in the sense of vector-valued distributions. Next we summarize some of the properties of the spectrum of functions of subexponential growth. Proposition 0.4 Let f, g ∈ L1loc (⺢; X) be of subexponential growth, α ∈ ⺓ \ {0}, K ∈ BV (⺢; B(X; Z)) with supp dK compact. Then (i) σ(f ) is closed; (ii) σ(αf ) = σ(f ); (iii) σ(f (· + h)) = σ(f ); (iv) σ(f˙) ⊂ σ(f ); (v) σ(f + g) ⊂ σ(f ) ∪ σ(g); (vi) σ(dK ∗ f ) ⊂ σ(f ); ) ⊂ σ(f ) − supp dK. (vii) σ(dKf Proof: (i) and (ii) are trivial by the definition of σ(f ). (iii) follows from the identity h λh ˆ ˆ f (· + h)(λ) = e f (λ) − eλ(h−t) f (t)dt,

Re λ = 0,

0

(0.51)

since the second term on the right is an entire function. (iv) If f˙ ∈ L1loc (⺢; X) is of subexponential growth, then f˙(λ) = λfˆ(λ) − f (0),

Re λ = 0;

this implies (iv). (v) If ρ ∈ σ(f ) ∪ σ(g) then fˆ(λ) and gˆ(λ) admit analytic extension to a ball Bε (iρ0 ), hence ρ ∈ σ(f + g) . (vi) Since supp dK is compact, i.e. K(t) is constant for t ≥ N , t ≤ −N , say, dK ∗ f is well-defined and ∞ N ∞ e−ε|t| |dK ∗ f |dt ≤ |dK(τ )|( e−ε|t+τ | |f (t)|dt) −∞

−N N

≤ (

−N

−∞

|dK(τ )|e

ε|τ |

∞

) −∞

e−ε|t| |f (t)|dt,

hence dK ∗ f is again of subexponential growth. A simple computation yields for Re λ = 0 ∞ ∞ τ (dK ∗ f )(λ) = ( dK(τ )e−λτ )fˆ(λ) + dK(τ ) f (t − τ )e−λt dt, (0.52) −∞

−∞

0

and so (vi) follows since the coefficient of fˆ(λ) and the second term are entire.

Preliminaries

21

(vii) We have for Re λ > 0 · f )(λ) (dK

∞

dK(t)f (t)e−λt dt =

=

0

∞ −∞

∞

∞

dK(ρ)

f (t)e−(λ+iρ)t dtds

0

dK(ρ)fˆ(λ + iρ)dρ,

=

(0.53)

−∞

and similarly for Re λ < 0. This formula defines an analytic extension to all λ = iρ0 , such that ρ0 + ρ ∈ σ(f ), for ρ ∈ supp dK i.e. (vii) holds. 2

The spectrum of a function also enjoys a certain continuity property which turns out to be very useful. Theorem 0.8 Let fn , f ∈ L1loc (⺢; X) be of subexponential growth and such that ∞ −ε|t| (i) −∞ e |fn (t) − f (t)|dt → 0 as n → ∞, for each ε > 0; (ii) there are c > 0 and k ∈ ⺞0 such that −k |f , n (λ)| ≤ c|Re λ|

0 < |Re λ| < δ, n ∈ ⺞.

¯ equivalently Then σ(fn ) ⊂ Λ for all n ∈ ⺞ implies σ(f ) ⊂ Λ; σ(f ) ⊂ ∩m≥1 ∪n≥m σ(fn ).

(0.54)

ˆ Proof: (i) implies f n (λ) → f (λ) uniformly on compact subsets of ⺓ \ i ⺢. Let Λ ⊂ ⺢ be closed and such that σ(fn ) ⊂ Λ for all n ∈ ⺞. Choose ρ0 ∈ Λ, and let r ≤dist(ρ0 , Λ)/4, r ≤ δ/2. We want to prove that fˆ(λ) admits analytic extension to Bε (iρ0 ); for this it sufficient to prove |f n (λ)| ≤ M

for all n ∈ ⺞, |λ − iρ0 | ≤ r,

(0.55)

by Montel’s theorem (cp. Conway [59]). For this purpose observe that f n (λ) is holomorphic on Bε (iρ0 ), hence Cauchy’s theorem yields with h(λ) = [(λ − iρ0 )(1 +

(λ − iρ0 )2 k )] 4r2

the relation h(λ)f n (λ) =

1 2πi

|z−iρ0 |=2r

h(z)f n (z)

dz , z−λ

λ ∈ Br (iρ0 ), n ∈ ⺞.

Since for |z − iρ0 | = 2r we have |h(z)| = 2k | Re z|k , (ii) yields |h(λ)f n (λ)| ≤

c 2π

|z−iρ0 |=2r

2k | Re z|k · | Re z|−k

|dz| ≤ 2k+1 c, |z − λ|

22

Preliminaries

for all λ ∈ Br (iρ0 ), n ∈ ⺞, and so sup λ∈Br (iρ0 )

|f n (λ)|

=

sup |λ−iρ0 |=r

k+1 |f n (λ)| ≤ c2

sup |λ−iρ0 |=r

|

1 | h(λ)

4 ≤ 2k+1 c( )k r−k = M, 3 independently of n ∈ ⺞, i.e. (0.55) follows and the proof is complete. 2

Observe that (i) of Theorem 0.8 is implied by |fn | ≤ g

a.e., fn → f a.e.

(0.56)

where g ∈ L1loc (⺢) is of subexponential growth, while (ii) follows if g is growing polynomially, i.e. ∞ |g(t)|(1 + |t|)−m dt < ∞ for some m ≥ 0. −∞

In fact, then |f n (λ)|

≤

∞

g(t)e 0

≤ (

∞

−Reλt

dt ≤ ( −m

g(t)(1 + t) 0

∞ 0

g(t)(1 + t)−m dt) · sup[(1 + t)m e−Reλt ]

m m 1−m dt)( ) e , Re λ

t>0

0 < Re λ < 1,

and similarly for 0 > Re λ > −1. 0.6 The Spectrum of Functions Growing Polynomially We have seen in Example 0.1, (ii) that σ(f ) = suppf˜ holds for f ∈ L1 (⺢). This characterization of the spectrum allows for a considerable extension if the Fourier transform is understood in the sense of distributions. For this purpose let f ∈ L1loc (⺢; X) be of polynomial growth; the tempered distribution Df corresponding to f is then defined according to ∞ [Df , ϕ] = f (t)ϕ(t)dt, ϕ ∈ S, (0.57) −∞

where S denotes the Schwartz space of all C ∞ -functions on ⺢ with each of its derivatives decaying faster than any polynomial. The Fourier transform of Df is defined by f , ϕ] = [Df , ϕ], [D ˜ ϕ ∈ S. (0.58) Recall also the definition of supp Df . A number ρ ∈ ⺢ belongs to supp Df if for every ε > 0 there is ϕ ∈ S such that supp ϕ ⊂ Bε (ρ) and [Df , ϕ] = 0. Proposition 0.5 Let f ∈ L1loc (⺢; X) be of polynomial growth. Then f ; (i) σ(f ) = suppD

Preliminaries

23

(ii) σ(f ) = ∅ ⇐⇒ f ≡ 0; n (iii) σ(f ) = {ρ1 , . . . , ρn } ⇐⇒ f (t) = k=1 pk (t)eiρk t , for some polynomials pk (t). (iv) σ(f ) compact ⇐⇒ f admits extension to an entire function of exponential growth. Proof: (i) If ρ0 ∈ σ(f ) then fˆ(λ) admits holomorphic extension to some ball Bε (iρ0 ). Let ϕ ∈ S be such that supp ϕ ⊂ Bε (ρ0 ); then ∞ ∞ ∞ (fˆ(iρ + σ) − fˆ(iρ − σ))ϕ(ρ)dρ = ( f (t)e−iρt e−σ|t| dt)ϕ(ρ)dρ −∞ −∞ −∞ ∞ = f (t)e−σ|t| ϕ(t)dt ˜ −∞

implies with σ → 0+ f , ϕ] [D

= =

∞

∞

f (t)ϕ(t)dt ˜ = lim f (t)e−σ|t| ϕ(t)dt ˜ σ→0+ −∞ −∞ ∞ (fˆ(iρ + σ) − fˆ(iρ − σ))ϕ(ρ)dρ = 0, lim

σ→0+

−∞

f . This shows the inclusion suppD f ⊂ σ(f ). i.e. ρ0 ∈ suppD To prove the converse inclusion we use Theorem 0.8. For this purpose let ∞ ψ ∈ C0∞ (⺢) be such that supp ψ ⊂ (−1, 1) , ψ ≥ 0, and −∞ ψ(ρ)dρ = 1; define n → 1 as n → ∞, uniformly for bounded t, and ψ n ∈ S. ψn (ρ) = nψ(nρ). Then ψ 1 Since f is polynomially growing, fn = ψn f belongs to L (⺢; X), fn → f uniformly for bounded t, and |fn | ≤ |f | = g. Thus (0.56) holds, which implies (i) and (ii) of Theorem 0.8 since f is growing polynomially. Example 0.1, (ii) yields σ(fn ) = suppf n ⊂ Bε (suppDf )

for n ≥ n(ε).

In fact, if ϕ ∈ S is such that supp ϕ ⊂ Bε/2 (ρ0 ), then ∞ ∞ ∞ n (t)ϕ f (ρ)ϕ(ρ)dρ = f (t) ϕ(t)dt = f (t)ψ n (t)dt n n −∞ −∞ −∞ ∞ f , ψn ∗ ϕ]; f (t)ψn ∗ ϕ(t)dt = [D = −∞

with supp (ψn ∗ ϕ) ⊂ supp ϕ + supp ψn ⊂ Bε/2 (ρ0 ) + (−1/n, 1/n) we conclude f ) ≥ ε and ε/2 > 1/n. Therefore, by Theorem f , ψn ∗ ϕ] = 0 if dist (ρ0 , supp D [D f . 0.8 we obtain σ(f ) ⊂ Bε (supp Df ) for every ε > 0, i.e. σ(f ) ⊂ supp D f = 0 by (i), hence Df = 0, and so f = 0. (ii) σ(f ) = ∅ implies D (iii) Consider first the special case σ(f ) = {0}; we then have to show that f is a polynomial. For this purpose consider the Laurent expansion of fˆ(λ) at λ = 0 fˆ(λ) =

∞ −∞

an λn ,

λ ∈ ⺓ \ {0}.

24

Preliminaries

The coefficients an are given by 1 fˆ(z)z −n−1 dz, an = 2πi |z|=r

n ∈ ⺪,

where r > 0 is arbitrary. Since f is of polynomial growth by assumption, there follows the estimate |fˆ(λ)| ≤ c|Re λ|−k ,

0 < |Re λ| < 1,

for some k ∈ ⺞0 . This implies z2 1 z n+k−1 (1 + 2 )k fˆ(z)dz| ≤ c2k rn , | 2πi |z|=r r

(0.59)

for all n ∈ ⺪, r > 0,

hence c2k rn ≥ |

k k l=0

l

r−2l ·

1 2πi

|z|=r

z n+k−1+2l fˆ(z)dz| = |

k k l=0

l

r−2l a−n−k−2l |.

Multiplying with r2k and letting r → 0 then implies a−n−k = 0 for all n ≥ 1, i.e. k λ = 0 is a pole of order at most k of fˆ(λ). Let f0 (t) = l=0 a−l−1 tl /l! ; obviously −1 f0 (λ) = −k al λ−l and so fˆ(λ) − f0 (λ) is entire, i.e. σ(f − f0 ) = ∅, hence f = f0 . The general case can now easily be reduced to this special case. (iv) Suppose σ(f ) is compact; we then define −1 f0 (z) = fˆ(λ)eλz dλ, z ∈ ⺓, 2πi Γ where Γ denotes any closed rectifiable simple contour surrounding σ(f ) clockwise. It is easy to verify that for any |Re λ| > maxμ∈Γ |Re μ| the identity 1 dμ f0 (λ) = fˆ(μ) 2πi Γ μ−λ holds. Thus it remains to show that the right hand side of this equation equals fˆ(λ). For this purpose let R > 0 be sufficiently large, i.e. R > |λ|, and R > max{|ρ| : ρ ∈ σ(f )}. Then Cauchy’s theorem yields the identity 1 dμ μ2 λ2 k ˆ (1 + 2 )k fˆ(μ) (1 + 2 ) f (λ) = (0.60) R 2πi Γ∪ΓR R μ−λ + where Γ = Γ− ∪ Γ+ , ΓR = Γ− R ∪ ΓR (cp. Fig. 0.1) and k from (0.59). Similarly to the estimate in step (iii) of this proof, we obtain 1 μ2 dμ 1 | ≤ c2k R−k sup → 0 as R → ∞, (1 + 2 )k fˆ(μ) | 2πi ΓR R μ−λ |μ − λ| |λ|=R

hence with R → ∞, (0.60) implies f0 (λ) = fˆ(λ) for all Re λ = 0. By uniqueness of the Laplace transform this yields f0 ≡ f . The converse follows from the Taylor series of entire functions. 2

Preliminaries

25

Γ σ(f)

-R

λ

0

Γ-

R Γ+

Figure 0.1: Integration path for (0.60)

f allows for some further useful properties. The characterization σ(f ) = supp D Proposition 0.6 Let f ∈ L1loc (⺢; X) be of polynomial growth, and let ψ ∈ S, the Schwartz space. Then (i) σ(ψ ∗ f ) ⊂ σ(f ) ∩ supp ψ; (ii) σ(f − ψ ∗ f ) ⊂ σ(f ) ∩ supp(1 − ψ); (iii) ψ ∗ f = f if ψ = 1 on a neighbourhood of σ(f ); (iv) σ(f ) ∩ supp ψ = ∅ implies ψ ∗ f = 0. Proof: Observe first that ψ ∗ f is well-defined and of polynomial growth again when f is of this class and ψ belongs to S. The principal property which leads to Proposition 0.6 is the identity ˜ ψ∗f , ϕ] = [D f , ψϕ], [D

ϕ ∈ S.

(0.61)

To prove (i), suppose ρ ∈ σ(f ); then σ(f ) ∩ B ε (ρ) = ∅ for some ε > 0. Let ϕ ∈ ˜ = 0, ψ∗f , ϕ] = [D f , ψϕ] C0∞ (⺢) such that supp ϕ ⊂ B ε (ρ); with (0.61) we obtain [D ˜ since supp ψϕ ⊂ B ε (ρ), by the definition of supp Df . Therefore σ(ψ ∗ f ) ⊂ σ(f ). ˜ then supp ψ˜ ∩ B ε (ρ) = ∅ for some On the other hand, suppose ρ ∈ supp ψ; ∞ ψ∗f , ϕ] = ε > 0. Let ϕ ∈ C0 (⺢), supp ϕ ⊂ B ε (ρ); (0.61) yields this time [D ˜ ˜ ˜ ˜ [Df , ψϕ] = 0, since supp ψϕ = supp ψ∩ supp ϕ = ∅, i.e. ψϕ = 0. Therefore σ(ψ ∗ f ) ⊂ supp ψ˜ follows. To prove (ii) observe that (i) and Proposition 0.4 already imply σ(f − ψ ∗ f ) ⊂ ˜ and choose ε > 0 such that σ(f ) ∪ σ(ψ ∗ f ) ⊂ σ(f ). So let ρ ∈ supp (1 − ψ), ∞ ˜ supp (1 − ψ) ∩ B ε (ρ) = ∅. Then for ϕ ∈ C0 (⺢), with supp ϕ ⊂ B ε (ρ); we have ˜ ˜ = 0. Therefore σ(f − ψ ∗ f ) ⊂ f , (1 − ψ)ϕ] f −ψ∗f , ϕ] = [D = 0, since (1 − ψ)ϕ [D ˜ supp (1 − ψ), and (ii) follows. The remaining assertions are direct consequences of (i) and (ii). 2

26

Preliminaries

The last result of this subsection is useful in detecting the spectrum of bounded functions, in particular of almost periodic and asymptotically almost periodic functions. Proposition 0.7 Let f ∈ L∞ (⺢; X) and ρ ∈ σ(f ). Then there is a constant M (ρ) > 0 such that |

t

0

e−iρs f (s)ds| ≤ M (ρ),

for all t > 0.

In particular α(ρ, f ) := lim N −1 N →∞

N 0

e−iρs f (s)ds = 0

for all ρ ∈ σ(f ).

Proof: Replacing f (t) by e−iρt f (t) if necessary, we may assume ρ = 0. If 0 ∈ σ(f ) then fˆ(λ) is holomorphic and bounded on some ball B2r (0). Let Γr denote the circle |λ| = r oriented counterclockwise and Γ+ , Γ− r its parts located in the t −λs r right resp. left halfplane. Define gt (λ) = 0 e f (s)ds, where t > 0 is fixed but arbitrary otherwise; gt (λ) is an entire function, hence Cauchy’s theorem yields 1 λ2 dλ gt (0) = gt (λ)eλt (1 + 2 ) . 2πi Γr r λ Similarly 1 fˆ(0) = 2πi

λ2 dλ fˆ(λ)eλt (1 + 2 ) , r λ Γr

by holomorphy of fˆ(λ) in B2r (0). Subtracting and collecting terms we obtain

t

f (s)ds 0

= gt (0) = fˆ(0) − fˆ(λ)ht (λ)dλ Γ− r + (gt (λ) − fˆ(λ))ht (λ)dλ + Γ+ r

where

Γ− r

gt (λ)ht (λ)dλ,

ht (λ) = (2πiλ)−1 eλt (1 + λ2 /r2 ).

Since fˆ(λ) is bounded on Γr , say by M , and ht (λ) is so on Γ− r , the first integral is bounded by M . For the second one we have ∞ ˆ |gt (λ) − f (λ)| ≤ |f (τ )| exp(−Re λτ )dτ ≤ |f |∞ · exp(−Re λt)/ Re λ, t

for all Re λ > 0, and |ht (λ)| ≤ (2πr)−1 exp(Re λt)(2Re λ)/r,

Preliminaries

27

therfore the second integral is estimated by |f |∞ /r. For the last one we obtain similarly t −1 |gt (λ)ht (λ)| ≤ (2πr) exp(Re λt)(2|Re λ|)/r · |f (τ )| exp(−Re λτ )dτ 0

≤ |f |∞ /πr2 , hence the integral is dominated also by |f |∞ /r. Summing up there follows |

t 0

f (τ )dτ | ≤ 2(M + |f |∞ /r),

hence the result is proved. 2

t > 0,

Chapter I

Equations of Scalar Type

1

Resolvents

The concept of the resolvent which is central for the theory of linear Volterra equations is introduced and discussed. It is applied to the inhomogeneous equation to derive various variation of parameters formulas. The main tools for proving existence theorems for the resolvent are described in detail; these methods are the operational calculus in Hilbert spaces, perturbation arguments, and the Laplace-transform method. The generation theorem, the analog of the Hille-Yosida theorem of semigroup theory for Volterra equations, proved in Section 1.5, is of fundamental importance in later chapters. The theory is completed with several counterexamples, and with a discussion of the integral resolvent. 1.1 Well-posedness and Resolvents Let X be a complex Banach space, A a closed linear unbounded operator in X with dense domain D(A), and a ∈ L1loc (⺢+ ) a scalar kernel ≡ 0. We consider the Volterra equation t u(t) = f (t) + a(t − s)Au(s)ds, t ∈ J, (1.1) 0

where f ∈ C(J; X), J = [0, T ]. In the sequel we denote by XA the domain of A equipped with the graph norm | · |A of A, i.e. |x|A = |x| + |Ax|; XA is a Banach space since A is closed, and it is continuously and densely embedded into X. We shall use the abbreviation t (a ∗ f )(t) = a(t − s)f (s)ds, t ∈ J, 0

for the convolution. The following notions of solutions of (1.1) are natural. Definition 1.1 A function u ∈ C(J; X) is called (a) strong solution of (1.1) on J if u ∈ C(J; XA ) and (1.1) holds on J; (b) mild solution of (1.1) on J if a ∗ u ∈ C(J; XA ) and u(t) = f (t) + A(a ∗ u)(t) on J; (c) weak solution of (1.1) on J if < u(t), x∗ > = < f (t), x∗ > + < (a ∗ u)(t), A∗ x∗ > on J, for each x∗ ∈ D(A∗ ). Obviously, every strong solution of (1.1) is a mild solution, and each mild solution is a weak one. If in addition, the resolvent set ρ(A) of A is nonempty, then every weak solution is also a mild one; see Proposition 1.4 in Section 1.2. However, in general not every mild solution is a strong solution. Note that in case a(t) ≡ 1 and f ∈ C 1 (J; X), (1.1) is equivalent to the Cauchy problem u(t) ˙ = f˙(t) + Au(t), u(0) = f (0). (1.2) J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_1, © Springer Basel 1993

30

1. Resolvents

31

A moment of reflection shows that our solution concepts coincide with those which are usually employed with (1.2). Similarly, if a(t) ≡ t and f ∈ C 2 (J; X), (1.1) is equivalent to u ¨(t) = f¨(t) + Au(t),

u(0) = f (0), u(0) ˙ = f˙(0),

(1.3)

and our solution concepts for (1.1) become those for second order problems (1.3). There are two natural transformations associated with (1.1). The first one consists of an exponential shift which results from multiplication of (1.1) with e−ωt . Then uω (t) = u(t)e−ωt will be again a solution of (1.1) with f (t) replaced by fω (t) = f (t)e−ωt , and a(t) by aω (t) = a(t)e−ωt . This way the kernel can be made integrable whenever a(t) is of exponential growth. The second transformation concerns the operator A. Adding ωa ∗ u to both sides of (1.1) and defining v = u + ωa ∗ u, (1.1) transforms into an equation of the same form, but with A replaced by A + ω and u by v, and a by the resolvent kernel rω , i.e. the solution of r + ωa ∗ r = a. This way one can assume 0 ∈ ρ(A) whenever ρ(A) = ∅. These transformations can always be carried out when considering (1.1) locally on ⺢+ , however, they change asymptotic behaviour. The following definition of well-posedness is a direct extension of well-posed Cauchy problems (1.2). Definition 1.2 Equation (1.1) is called well-posed if for each x ∈ D(A) there is a unique strong solution u(t; x) on ⺢+ of u(t) = x + (a ∗ Au)(t),

t ≥ 0,

(1.4)

and (xn ) ⊂ D(A), xn → 0 imply u(t; xn ) → 0 in X, uniformly on compact intervals. Suppose (1.1) is well-posed; we then may introduce the solution operator S(t) for (1.1). S(t)x = u(t; x), x ∈ D(A), t ≥ 0. (1.5) By uniqueness, S(t) is unambigously defined and linear for each t ≥ 0, S(0)x = x holds on D(A), and S(t)x is continuous on ⺢+ , for each x ∈ D(A). We show that S(t) is also uniformly bounded on compact intervals. To see this, assume on the contrary there are sequences (tn ) ⊂ [0, T ] and (yn ) ⊂ D(A), |yn | = 1, such that |S(tn )yn | ≥ n for each n ∈ ⺞; then xn = yn /n ∈ D(A), xn → 0, hence from Definition 1.2 we obtain the contradiction 1 ≤ |S(tn )xn | = |u(tn ; xn )| → 0

as

n → ∞.

Thus S(t) is bounded and therefore admits extension to all of X, S(t)x is continuous for each x ∈ X. By definition of strong solutions it is further clear that

32

Chapter I. Equations of Scalar Type

the solution operator maps D(A) into itself, AS(t)x is continuous on ⺢+ for each x ∈ D(A) and S(t)x = x + a ∗ AS(t)x = x + Aa ∗ S(t)x holds on ⺢+ . Since S(t) is bounded we obtain further R(a ∗ S(t)) ⊂ D(A) and Aa ∗ S(t) = S(t) − I is strongly continuous on ⺢+ ; in other words u(t; x) = S(t)x is a mild solution of (1.4) for each x ∈ X. Next we observe that also the mild solutions of (1.4) are unique. In fact, if u = Aa ∗ u then v = a ∗ u is a strong solution of (1.4) with x = 0 , hence v = 0 by uniqueness of the strong solutions; Titchmarsh’s theorem (cp. e.g. Yosida [347], p. 166) then implies u = 0. Finally, we see that A commutes with S(t); in fact, for x ∈ D(A), both u(t; Ax) and Au(t; x) are mild solutions of (1.4) with x replaced by Ax, hence S(t)Ax = u(t; Ax) = Au(t; x) = AS(t)x

for all x ∈ D(A), t ≥ 0.

These considerations give rise to the following Definition 1.3 A family {S(t)}t≥0 ⊂ B(X) of bounded linear operators in X is called a resolvent for (1.1) [or solution operator for (1.1)] if the following conditions are satisfied. (S1) S(t) is strongly continuous on ⺢+ and S(0) = I; (S2) S(t) commutes with A, which means that S(t)D(A) ⊂ D(A) and AS(t)x = S(t)Ax for all x ∈ D(A) and t ≥ 0; (S3) the resolvent equation holds t S(t)x = x + a(t − s)AS(s)xds for all x ∈ D(A), t ≥ 0. (1.6) 0

We have proved so far that a well-posed problem (1.1) admits a resolvent; the converse is also true. Proposition 1.1 (1.1) is well-posed iff (1.1) admits a resolvent S(t). If this is the case we have in addition R(a ∗ S(t)) ⊂ D(A) for all t ≥ 0 and t a(t − s)S(s)xds for all x ∈ X, t ≥ 0. (1.7) S(t)x = x + A 0

In particular, Aa ∗ S is strongly continuous in X. Proof: The only thing left to prove is that existence of a resolvent S(t) implies wellposedness of (1.1). So suppose S(t) is a resolvent for (1.1). Then, by the properties of S(t), for each x ∈ D(A) the function u(t) = S(t)x is a strong solution of (1.4). To prove its uniqueness let v(t) be another strong solution of (1.4); by (S2) and (S3) as well as by (1.4) we obtain 1∗v

= (S − a ∗ AS) ∗ v = S ∗ v − S ∗ a ∗ Av = S ∗ (v − a ∗ Av) = S ∗ x = 1 ∗ Sx,

1. Resolvents

33

hence after differentiation v(t) = S(t)x. Continuous dependence of the solutions on x follows directly from uniform boundedness of S(t) on compact intervals which in turn is a consequence of (S1), by the uniform boundedness principle. 2

Corollary 1.1 (1.1) admits at most one resolvent S(t). Proof: If S1 (t) and S2 (t) are both resolvents for (1.1), then for x ∈ D(A) we obtain 1 ∗ S1 x = (S2 − a ∗ AS2 ) ∗ S1 x = S2 ∗ S1 x − a ∗ S2 ∗ AS1 x = S2 ∗ (S1 x − a ∗ AS1 x) = S2 ∗ x = 1 ∗ S2 x, hence S1 (t)x = S2 (t)x for each x ∈ D(A), t ≥ 0, and so S1 (t) = S2 (t) by density of D(A). 2

For the special cases a(t) = 1 and a(t) = t mentioned above the resolvent S(t) becomes the C0 -semigroup eAt generated by A, resp. the cosine family Co(t) generated by A. 1.2 Inhomogeneous Equations Suppose S(t) is a resolvent for (1.1) and let u(t) be a mild solution of (1.1). Then from (S1) ∼ (S3) we obtain 1 ∗ u = (S − A(a ∗ S)) ∗ u = S ∗ u − AS ∗ (a ∗ u) = S ∗ u − S ∗ (Aa ∗ u) = S ∗ (u − Aa ∗ u) = S ∗ f, i.e. S ∗ f is continuously differentiable and u(t) =

d dt

t 0

S(t − s)f (s)ds,

t ∈ J.

(1.8)

This is the variation of parameters formula for Volterra equations (1.1). Proposition 1.2 Suppose (1.1) admits a resolvent S(t) and let f ∈ C(J; X). Then (i) if u ∈ C(J; X) is a mild solution of (1.1), then S ∗ f is continuously differentiable on J and d t u(t) = S(t − s)f (s)ds, t ∈ J; (1.9) dt 0 in particular, mild solutions of (1.1) are unique, (ii) if f ∈ W 1,1 (J; X) then

t

u(t) = S(t)f (0) + 0

is a mild solution of (1.1),

S(t − s)f˙(s)ds,

t ∈ J,

(1.10)

34


(iii) if f = x + a ∗ g, with g ∈ W 1,1 (J; X) and x ∈ D(A), then u(t) = S(t)x + a ∗ S(t)g(0) + a ∗ S ∗ g(t), ˙

t ∈ J,

(1.11)

is a strong solution of (1.1); (iv) if f ∈ W 1,1 (J; XA ) then u(t) given by (1.10) is a strong solution of (1.1). Proof: (ii) If f ∈ W 1,1 (J; X) then u(t) defined by (1.10) is continuous on J and we obtain from Proposition 1.1 a ∗ u = a ∗ Sf (0) + a ∗ S ∗ f˙ ∈ D(A)

for each

t ∈ J,

and Aa ∗ u

= Aa ∗ Sf (0) + (Aa ∗ S) ∗ f˙ = (S − I)f (0) + S ∗ f˙ − 1 ∗ f˙ = Sf (0) + S ∗ f˙ − f (0) − 1 ∗ f˙ = u − f,

i.e. u(t) is a mild solution of (1.1). (iii) Let v = Sg(0) + S ∗ g; ˙ by (ii) v is a mild solution of (1.1) with f replaced by g, in particular we have a ∗ v ∈ C(J; XA ). Now, u(t) given by (1.11) becomes u = Sx + a ∗ v, hence u ∈ C(J; XA ) since x ∈ D(A) and so u is a strong solution of (1.1). (iv) This is a direct consequence of (ii) since S(t) commutes with A. 2

Suppose for a moment a ∈ BVloc (⺢+ ); then, given x ∈ D(A), we obtain from the 1,∞ resolvent equation (1.6) that S(·)x is of class Wloc (⺢+ ; X) and t ˙ S(t)x = da(s)S(t − s)Ax, t ≥ 0. (1.12) 0

In this case (1.8) can be rewritten as t ˙ − s)f (s)ds, u(t) = f (t) + S(t 0

t ∈ J,

(1.13)

and the right hand side of this equation makes sense if we only know f ∈ C(J; XA ) rather than f ∈ W 1,1 (J; XA ). In general, S(t) will not be differentiable on D(A), this leads to a restricted class of resolvents. Definition 1.4 A resolvent S(t) for (1.1) is called differentiable, if S(·)x ∈ 1,1 Wloc (⺢+ ; X) for each x ∈ D(A) and there is ϕ ∈ L1loc (⺢+ ) such that ˙ |S(t)x| ≤ ϕ(t)|x|A

a.e. on ⺢+ , for each x ∈ D(A).

(1.14)

We have shown above that S(t) is differentiable in case a(t) belongs to BVloc (⺢+ ). Whenever S(t) is differentiable then (1.13) makes sense for each f ∈ C(J; XA ) and from this we obtain.

1. Resolvents

35

Proposition 1.3 Suppose (1.1) admits a differentiable resolvent S(t). Then (i) if f ∈ C(J; XA ) then u(t) defined by (1.13) is a mild solution of (1.1); (ii) if f = x + a ∗ g, with g ∈ C(J; XA ) and x ∈ D(A), then u(t) = S(t)x + a ∗ g(t) + a ∗ S˙ ∗ g(t),

t ∈ J,

(1.15)

is a strong solution of (1.1). ˙ Proof: If S(t) is differentiable, then S(t) satisfies ˙ ˙ S(t)x = a(t)Ax + Aa ∗ S(t)x,

a.a. t ≥ 0, x ∈ D(A);

(1.16)

˙ in particular a ∗ S(t) maps D(A) into D(A) for a.a. t ≥ 0 and ˙ ˙ |Aa ∗ S(t)x| ≤ |S(t)x| + |a(t)Ax| ≤ (ϕ(t) + |a(t)|)|x|A . This implies for f ∈ C(J; XA ) that a ∗ u = a ∗ f + a ∗ S˙ ∗ f belongs to C(J; XA ), hence u is a mild solution of (1.1) by (1.16). If f = x + a ∗ g with g ∈ C(J; XA ) then v = g + S˙ ∗ g is a mild solution of (1.1) with f replaced by g, hence a ∗ v ∈ C(J; XA ). But (1.15) reads u = Sx + a ∗ v and so u ∈ C(J; XA ) since x ∈ D(A) and u is a strong solution of (1.1) by (1.16). 2

We conclude this section demonstrating that the concepts mild solution and weak solution for well-posed equations of the form (1.1) coincide since then ρ(A) = ∅, as we shall see in the next section. Proposition 1.4 Let f ∈ C(J; X) and suppose ρ(A) = ∅. Then each weak solution u on J of (1.1) is a mild solution of (1.1) on J. Proof: Suppose μ ∈ ρ(A) and let u ∈ C(J; X) be a weak solution of (1.1); then for each x∗ ∈ D(A∗ ) we have < a ∗ u, A∗ x∗ > = < u − f, x∗ > hence

on J,

< a ∗ u, (μ − A∗ )x∗ > = < f − u + μa ∗ u, x∗ > ∗

∗

∗

on J.

∗

Since μ − A is invertible, we obtain with y = (μ − A )x < a ∗ u, y ∗ > = =

< f − u + μa ∗ u, (μ − A∗ )−1 y ∗ > < (μ − A)−1 (f − u + μa ∗ u), y ∗ >

for all y ∗ ∈ X ∗ , hence a ∗ u = (μ − A)−1 (f − u + μa ∗ u) But this implies a ∗ u(t) ∈ D(A) for each t ∈ J and Aa ∗ u(t) = u(t) − f (t) i.e. u(t) is a mild solution of (1.1). 2

on J,

on J.

36


1.3 Necessary Conditions for Well-posedness Suppose S(t) is a resolvent for (1.1) and let −μ ∈ σ(A) be an eigenvalue of A with eigenvector x = 0. Then S(t)x = s(t; μ)x,

t ≥ 0,

(1.17)

where s(t; μ) is the solution of the one-dimensional Volterra equation

t

s(t; μ) + μ 0

a(t − τ )s(τ ; μ)dτ = 1,

t ≥ 0.

(1.18)

In fact, Aa ∗ s(t; μ)x = a ∗ s(t; μ)Ax = −μa ∗ s(t; μ)x = s(t; μ)x − x, hence (1.17) holds, by uniqueness of the solutions of (1.4). Similarly, if S(t) is differentiable then ˙ S(t)x = μr(t; μ)x, t ≥ 0, (1.19) where r(t; μ) is the solution of the one-dimensional equation r(t; μ) + μ 0

t

a(t − τ )r(τ ; μ)dτ = a(t),

t ≥ 0.

(1.20)

These observations lead to necessary conditions for well-posedness of (1.1). Proposition 1.5 Suppose (1.1) admits a resolvent S(t), and let s(t; μ) and r(t; μ) be defined as the solutions of the scalar Volterra equations (1.18) and (1.20). Then (i) there is a locally bounded lower semicontinuous function ψ(t) such that |s(t; μ)| ≤ ψ(t)

for all − μ ∈ σ(A) and t ≥ 0;

(1.21)

(ii) if S(t) is differentiable, then there is ϕ ∈ L1loc (⺢+ ) such that |r(t; μ)| ≤ ϕ(t)

for all − μ ∈ σ(A) and a.a. t ≥ 0.

(1.22)

In particular, there is a sequence (μn ) ⊂ ρ(A) such that |μn | → ∞. Proof: (i) Suppose −μ ∈ σ(A)\σr (A); then there are sequences (xn ) ⊂ D(A), |xn | = 1 and (yn ) ⊂ X, yn → 0 such that (μ + A)xn = yn . Set un (t) = S(t)xn − s(t; μ)xn ; then un (t) satisfies the equation un + μa ∗ un = a ∗ Syn , hence un = r(·; μ) ∗ Syn → 0

as n → ∞,

1. Resolvents

37

uniformly on compact intervals. Therefore |s(t; μ)| = |s(t; μ)xn | ≤ |un (t)| + |S(t)xn | ≤ |un (t)| + |S(t)|, i.e. with n → ∞

|s(t; μ)| ≤ |S(t)|

for all t ≥ 0.

On the other hand, if −μ ∈ σr (A) then there is x∗ ∈ D(A∗ ), |x∗ | = 1, such that A∗ x∗ = −μx∗ ; given x ∈ X, this implies by (1.7) < Sx, x∗ > = < x, x∗ > −μa∗ < Sx, x∗ >, i.e.

< Sx, x∗ > = s(·; μ) < x, x∗ >

on ⺢+ ,

by uniqueness. Thus we obtain |s(t; μ)| = sup |s(t; μ) < x, x∗ > | = sup | < S(t)x, x∗ > | ≤ |S(t)|; |x|≤1

|x|≤1

this means that ψ(t) = |S(t)| is the function we are looking for. It follows from the strong continuity of S(t) that ψ(t) is locally bounded and lower semicontinuous. (ii) is proved by a similar argument, the function ϕ(t) comes from the definition of a differentiable resolvent. Finally, we show there is a sequence (μn ) ⊂ ρ(A) such that |μn | → ∞. Assume on the contrary ρ(A) ∩ {μ ∈ ⺓ : |μ| ≥ R} = ∅; then we have by (i) |s(t; μ)| ≤ ψ(t),

for all t ≥ 0

and all

μ ∈ ⺓, |μ| ≥ R.

Since s(t; ·) is an entire function for each fixed t ≥ 0, the latter implies s(t; μ) ≡ s(t; 0) = 1 by Liouville’s theorem and so (1.18) shows a(t) = 0 a.e., in contradiction to the standing assumption a(t) ≡ 0. 2

If X is a Hilbert space and A is a normal operator then the operational calculus for such operators can be used to construct the resolvent S(t).

Theorem 1.1 Let A be a normal operator in the Hilbert space X. Then (1.1) is well-posed iff there is a locally bounded lower semicontinuous function ψ(t) such that |s(t; μ)| ≤ ψ(t) for all − μ ∈ σ(A) and t ≥ 0. (1.23) The resolvent S(t) is then given by S(t) = s(t; −μ)dE(μ),

t ≥ 0,

(1.24)

σ(A)

where E denotes the spectral measure of A. S(t) is differentiable iff there is ϕ ∈ L1loc (⺢+ ) such that |r(t; μ)| ≤ ϕ(t)

for all − μ ∈ σ(A) and a.a. t ≥ 0.

(1.25)

38


For the concept of the spectral measure of a normal operator and to its operational calculus we refer to Dunford and Schwartz [103]. The proof of Theorem 1.1 is a simple exercise to this operational calculus and is therefore omitted. An easy computation shows that for a(t) = 1 we have r(t; μ) = s(t; μ) = e−μt ,

t ≥ 0, μ ∈ ⺓,

i.e. (1.23) means in this case Re σ(A) ≤ ω < ∞, the spectrum of A must necessarily be contained in a left halfplane. On the other hand, for a(t) = t we obtain √ s(t; μ) = cos( μt),

√ √ r(t; μ) = sin( μt)/ μ, t ≥ 0, μ ∈ ⺓. In this case (1.23) and (1.25) reduce to Re σ(A) ≤ ω, i.e. σ(A) must necessarily be contained inside of the parabola p(y) = ω 2 − y 2 + 2iωy, y ∈ ⺢. In general, it turns out to be quite difficult to obtain estimates like (1.23) and (1.25) and although Theorem 1.1 deals only with really nice operators it is in general only of limited use. However, two cases are worthy of mention since they appear naturally in applications. Recall that a measure da(t) is said to be of positive type if

T

Re 0

(da ∗ ϕ)(t)ϕ(t)dt ≥ 0,

for each ϕ ∈ C(⺢+ , ⺓) and T > 0.

If da(t) is of subexponential growth, it is of positive type iff Re da(λ) ≥ 0 for all Re λ > 0, cp. Gripenberg, Londen and Staffans [156] or the original paper of Nohel and Shea [258]. Corollary 1.2 Suppose A is selfadjoint and negative semidefinite in the Hilbert space X.Then the following assertions are valid. (i) Let a ∈ BVloc (⺢+ ) be such that the measure da is of positive type. Then (1.1) admits a differentiable resolvent S(t) such that |S(t)| ≤ 1,

and

˙ |S(t)x| ≤ Var a|t0 |Ax|,

for all t ≥ 0, x ∈ D(A).

(ii) Let a ∈ L1loc (⺢+ ) be nonnegative and nonincreasing. Then (1.1) admits a resolvent S(t) which is selfadjoint and such that 0 ≤ (S(t)x, x) ≤ 1 for all x ∈ X. Proof: (i) Since A is selfadjoint and negative semidefinite we have σ(−A) ⊂ ⺢+ ; let μ ∈ ⺢+ be fixed. Differentiate (1.18) and multiply with s(t; μ); this yields ∂ ∂ |s(t; μ)|2 = 2 Re ( s(t; μ)s(t; μ)) = −2μ Re [da ∗ s(t; μ)s(t; μ)], ∂t ∂t hence |s(t; μ)|2 = 1 − 2μ Re

t 0

da ∗ s(τ ; μ)s(τ ; μ)dτ ≤ 1,

1. Resolvents

39

since μ ≥ 0 and da is of positive type. Hence (1.23) holds with ψ(t) ≡ 1, and therefore (1.25) with ϕ(t) =Var a|t0 since r = da ∗ s. (ii) If a(t) is nonnegative and nonincreasing then 0 ≤ s(t; μ) ≤ 1, for all t, μ ≥ 0; this is a special case of a result due to Friedman [120]. The assertion follows then from Theorem 1.1. 2

Observe that if a(t) is nonnegative, nonincreasing, then a(0+) < ∞ implies a ∈ BV (⺢+ ) and da of positive type; this exhibits a connection between (i) and (ii) of Corollary 1.2. In the sections containing applications we shall refer frequently to this corollary. 1.4 Perturbed Equations Consider the perturbed equation u(t) = f (t) + (a + a ∗ k) ∗ Au(t) + b ∗ u(t),

t∈J

(1.26)

where a(t), A, f (t) are as before and k, b ∈ L1loc (⺢+ ). The concepts of solutions, well-posedness and resolvents have obvious extension to the perturbed problem (1.26); note that a ∗ u ∈ C(J; XA ) if and only if (a + a ∗ k) ∗ u ∈ C(J; XA ). By means of the variation of parameters formula obtained in Section 1.2 it is possible to show that (1.26) is well-posed iff (1.1) has this property, provided both, b and k, are locally of bounded variation. Theorem 1.2 Suppose k, b ∈ BVloc (⺢+ ). Then (1.26) is well-posed iff (1.1) is well-posed. Also, the resolvent S1 (t) of (1.26) is differentiable iff the resolvent S(t) of (1.1) has this property. Proof: Suppose (1.1) admits a resolvent S(t). The resolvent S1 (t) must satisfy the equation S1 = I + Aa ∗ S1 + k ∗ Aa ∗ S1 + b ∗ S1 , (1.27) hence by the variation of parameters formula (1.8) it must be a solution of S1

d (S ∗ (I + b ∗ S1 + k ∗ Aa ∗ S1 )) dt = S + db ∗ S ∗ S1 + dk ∗ (Aa ∗ S) ∗ S1 =

= S + [(db + dk) ∗ S − k] ∗ S1 , since b, k are of locally bounded variation. Thus S1 must be a solution of S1 = S + K ∗ S1 ,

(1.28)

0 where 1 ∗ K = (b + k) ∗ S − 1 ∗ k belongs to BVloc (⺢+ ; B(X)). Therefore, by Theorem 0.5, (1.28) has a unique strongly continuous solution S1 (t) on ⺢+ . Convolving (1.28) with a(t) we see that a ∗ S1 (t) maps into D(A), and applying A we obtain strong continuity of Aa ∗ S1 (t), since Aa ∗ S(t) has this property; note that

40


K commutes with A. For the same reason S1 (t) commutes with A and it satisfies (1.27) by construction and by Proposition 1.2. If S(t) is also differentiable, differentiation of (1.28) yields ˙ + K ∗ S˙ 1 x + Kx, S˙ 1 x = Sx hence S1 has also this property. The converse implications follow reversing the roles of S(t) and S1 (t). 2

1,1 Corollary 1.3 Suppose k, b ∈ Wloc (⺢+ ) and let S(t) be a resolvent for (1.1). Then the resolvent S1 (t) for (1.26) admits the decomposition

S1 (t) = S(t) + S2 (t),

t ≥ 0,

where S2 (t) is continuous in B(X) for all t ≥ 0. If S(t) is also differentiable then S˙ 2 ∈ L1loc (⺢+ , B(XA , X)). This corollary is simply due to the fact that for a strongly continuous (resp. strongly locally integrable) family of operators T (t) and a scalar kernel c ∈ L1loc (⺢+ ) the convolution c ∗ T is continuous in B(X) (resp. locally integrable in B(X)). For the special case a(t) = 1, (1.26) is equivalent to u(t) ˙ =

f˙(t) + Au(t) + k ∗ Au(t) + db ∗ u(t),

u(0) =

f (0);

t ≥ 0, (1.29)

Theorem 1.2 then states that (1.29) is well-posed iff A generates a C0 -semigroup in X. This observation yields Corollary 1.4 Suppose k, b ∈ BVloc (⺢+ ). Then (1.29) is well-posed iff A generates a C0 -semigroup in X. If this in the case, the resolvent for (1.29) is differentiable. On the other hand, for a(t) = t, (1.26) is equivalent to ˙ u ¨ = f˙ + Au + k ∗ Au + b(t)f (0) + db ∗ u, ˙ t ≥ 0, ˙ u(0) = f (0), u(0) ˙ = f (0) + b(0+)f (0);

(1.30)

in this case Theorem 1.2 states that (1.30) is well-posed iff A generates a cosine 1,1 family in X, provided k ∈ BVloc (⺢+ ) and b ∈ Wloc (⺢+ ). 1,1 Corollary 1.5 Suppose b ∈ Wloc (⺢+ ) and k ∈ BVloc (⺢+ ). Then (1.30) is wellposed iff A generates a cosine family in X. If this is the case, the resolvent for (1.30) is differentiable.

We conclude this subsection with examples showing that the assumptions of Theorem 1.2 cannot be relaxed considerably.

1. Resolvents

41

Example 1.1 Consider the problem u(t) ˙ = Au(t), t ≤ 1, u(t) ˙ = Au(t) + Au(t − 1), u(0)

t > 1,

(1.31)

= x;

(1.31) is of the form (1.26) with b(t) ≡ 0 and k(t) = δ0 (t − 1), a(t) ≡ 1, i.e. it is of the form (1.29). We show that (1.31) admits a resolvent S(t) iff A generates an analytic C0 -semigroup in X. In fact, suppose (1.31) admits a resolvent S(t); (1.31) for t ≤ 1 then implies that A must generate a C0 -semigroup eAt in X and S(t) = eAt holds for t ≤ 1. The formula of variation of parameters yields an explicit representation of S(t) for all t ≥ 0, namely S(t) =

∞ 1 (t − n)n+ An eA(t−n)+ , n! n=0

t ≥ 0,

(1.32)

where (τ )+ = max(τ, 0) for each τ ∈ ⺢. Let t = 1 + τ, τ ∈ (0, 1); (1.32) yields the identity τ AeAτ = S(1 + τ ) − eA(1+τ ) , τ ∈ (0, 1), hence |τ AeAτ | ≤ M for τ ≤ 1. But this estimate is well-known to be equivalent to analyticity of eAt . Conversely, if eAt is analytic it is easy to verify that formula (1.32) defines the resolvent for (1.31). 2

Example 1.2 Consider the initial-boundary value problem t ut (t, x) = ux (t, x) + k(t − τ )ux (τ, x)dτ, t ≥ 0, x ∈ [0, 2π], 0

u(0, x) = u0 (x), u(t, 0) = u(t, 2π),

x ∈ [0, 2π],

(1.33)

t ≥ 0.

Let X = L2 (0, 2π), Au = ux with D(A) = {u ∈ W 1,2 (0, 2π), u(0) = u(1)}; it is well-known that A is a normal, even skewadjoint operator in X. We will show that (1.33) does not admit a resolvent in X in case the kernel k(t) is chosen as k(t) = tα−1 /Γ(α),

t > 0, α ∈ (0, 1);

(1.34)

note that k(t) is even analytic for t > 0 but has a singularity at t = 0, it is not of bounded variation near t = 0. Since σ(A) = {in : n ∈ ⺪}, by Theorem 1.1 we have to study the functions sn (t) = s(t; −in), i.e. the solutions of s˙ n = insn + ink ∗ sn , sn (0) = 1,

t ≥ 0, n ∈ ⺪.

(1.35)

It suffices to show that {sn (t) : n ∈ ⺪} is not locally uniformly bounded, by Theorem 1.1. (i) The Laplace transform sn (λ) of sn is easily seen to be sn = (λ − in − in/λα )−1 ,

λ ∈ ⺓+ ;

42


s n (λ) clearly has essential singularities for λ ≤ 0, but it also has poles in ⺓\⺢− . To obtain the asymptotic behaviour of these poles we study the zeros of hn (λ) = λ − in − in/λα as n → ±∞. Rescaling by z = λ/|n|, these are the solutions of the equation z 1+α = (iz α + i|n|−α )sgn n. 1 z∞

(1.36)

2 z∞

As n → ±∞ they are given by = ±i and = 0, hence for |n| large, there is always a solution zn of (1.36) close to ±i. But this means |n|α (zn ∓ i) = ±i/znα → (±i)1−α as n → ±∞, i.e. λn = |n|zn ∼ in + (in)1−α as n → ±∞; note that Re λn = Re (in)1−α = |n|1−α cos( π2 (1 − α)) → ∞ as |n| → ∞. (ii) The complex inversion formula for the Laplace transform yields ωn +i∞ sn (t) = (2πi)−1 eλt (λ − in − inλ−α )−1 dλ, ωn −i∞

where ωn has to be chosen sufficiently large, in particular ωn > Re λn . Since for large |n| the functions hn have no zeros strictly away from the set ⺢− other than λn we may deform the integration path into Γ, which consists of the ray (∞, R]e−iϕ , the arc Rei[−ϕ,ϕ] and the ray [R, ∞)eiϕ . By Cauchy’s theorem this yields sn (t) = an eλn t + (2πi)−1

Γ

eλt (λ − in − inλ−α )−1 dλ,

(1.37)

where by residue calculus an = Res (λ − in − inλ−α )−1 |λ=λn = (1 + inα/λ1+α ) → 1, n

as n → ∞.

π The integral can be estimated by ceRt /|n|, provided ϕ ∈ ( π2 , 2α ) and R is large. Thus |sn (t)| ∼ |an |eReλn t ∼ eReλn t → ∞ as |n| → ∞,

i.e. (1.23) is violated and so (1.33) does not admit a resolvent. 2

1.5 The Generation Theorem Since (1.1) is a convolution equation on the halfline it is natural to employ the Laplace transform for its study. Besides the standing assumptions on a(t) and A we therefore suppose that a(t) is Laplace transformable, i.e. there is ω ∈ ⺢ such ∞ that 0 e−ωt |a(t)|dt < ∞. But we also have to restrict the class of resolvents, they must be Laplace transformable as well. Definition 1.5 Suppose S(t) is a resolvent for (1.1). (i) S(t) is called exponentially bounded if there are constants M ≥ 1 and ω ∈ ⺢ such that |S(t)| ≤ M eωt , for all t ≥ 0; (1.38) ω or more precisely (M, ω) is called a type of S(t).

1. Resolvents

43

(ii) The growth bound ω0 (S) of a resolvent S(t) for (1.1) is defined by ω0 (S) = limt→∞ t−1 log |S(t)|.

(1.39)

Obviously, we have the relation ω0 (S) = inf{ω : ∃M > 0 such that (1.38) holds }. Now, suppose S(t) is an exponentially bounded resolvent for (1.1); its Laplace transform ∞ H(λ) = S(λ) = e−λt S(t)dt, Re λ > ω, 0

is then well-defined and holomorphic for Re λ > ω, and satisfies the estimates |H (n) (λ)| ≤ M n! (Re λ − ω)−(n+1) ,

Re λ > ω, n ∈ ⺞0 .

(1.40)

To compute H(λ) we use the resolvent equations (1.6) and (1.7); in fact, we obtain from the convolution theorem for Re λ > ω the relation H(λ)x = x/λ + a(λ)H(λ)Ax, valid for each x ∈ D(A), and H(λ)x = x/λ + A a(λ)H(λ)x, for each x ∈ X. Thus the operators I − a(λ)A are bijective, i.e. invertible, and we obtain H(λ) = (I − a(λ)A)−1 /λ for Re λ > ω. In particular, 1/ a(λ) ∈ ρ(A) for all such λ, provided a(λ) = 0. Assume a(λ0 ) = 0 for some λ0 with Re λ0 > ω; since a(λ) is holomorphic, λ0 is an isolated zero of finite multiplicity, and H(λ0 ) = I/λ0 . Choose a small circle Γ around λ0 which is entirely contained in the halfplane Re λ > ω such that a(λ) = 0 on Γ; then AH(λ) = (H(λ) − I/λ)/ a(λ) is well-defined and holomorphic on Γ, hence by Cauchy’s formula we obtain λH(λ) λH(λ) − I −1 −1 A = Aλ0 H(λ0 ) = A(2πi) dλ = (2πi) dλ, λ − λ (λ − λ0 ) a(λ) 0 Γ Γ and so A is bounded, a contradiction to our standing hypotheses. Thus a(λ) = 0 for Re λ > ω. It turns out that these properties of a(λ) and A in combination with (1.40) characterize resolvents of type (M, ω). Theorem 1.3 Let A be a closed linear unbounded operator in X with dense ∞ domain D(A) and let a ∈ L1loc (⺢+ ) satisfy 0 e−ωt |a(t)|dt < ∞. Then (1.1) admits a resolvent S(t) of type (M, ω) iff the following conditions hold.

44


(H1) a(λ) = 0 and 1/ a(λ) ∈ ρ(A) for all λ > ω; (H2) H(λ) = (I − a(λ)A)−1 /λ satisfies the estimates |H (n) (λ)| ≤ M n!(λ − ω)−(n+1) ,

λ > ω, n ∈ ⺞0 .

(1.41)

Proof: It remains to prove sufficiency of (H1) and (H2). By Theorem 0.2 there is ω (λ) = H(λ + ω/λ, λ > 0. Define a Lipschitz family {Uω (t)}t≥0 ⊂ B(X) with U {U (t)}t≥0 ⊂ B(X) by t ωt U (t) = e Uω (t) − eωs Uω (s)ds, t ≥ 0; 0

then U (t) is locally Lipschitz and ˆ (λ) = H(λ)/λ, U

λ > ω.

The definition of H(λ) shows that U (t) commutes with A and yields the identity (λ) = λ−2 + (λ)A, U a(λ)U i.e. U (t)x = tx + a ∗ U (t)Ax

for all x ∈ D(A), t ≥ 0.

Since for each x ∈ D(A), f (t) = U (t)Ax is locally lipschitz, i.e. f ∈ BVloc (⺢+ ; X), 1,∞ we see that U (t)x belongs to Wloc (⺢+ ; X) and (U (t)x)· = x + (a ∗ f (t))· = x + a ∗ df (t),

for a.a. t ≥ 0.

But this implies a ∗ df (t) = a ∗ (U (t)Ax)· for all x ∈ D(A2 ) and so a ∗ df (t) is even continuous, from which in turn we see that U (t)x is continuously differentiable on ⺢+ for each x ∈ D(A2 ). D(A2 ) is dense in X and the operators (U (t + h) − U (t))/h are uniformly bounded for 0 < h ≤ 1, t bounded, hence by the BanachSteinhaus Theorem U (t)x is continuously differentiable on ⺢+ for each x ∈ X. Define S(t)x = U˙ (t)x, t ≥ 0, x ∈ X; it is clear that S(t)x is strongly continuous, of type (M, ω) and S(t) commutes with A since U (t) does and A is closed. Finally, S(λ) = H(λ) for Re λ > ω and by (H2) we obtain S(λ) = λ−1 + a(λ)S(λ)A, i.e. the resolvent equation (1.6) is satisfied. The proof is complete. 2

For the special case a(t) = 1, Theorem 1.3 becomes the celebrated generation theorem for C0 -semigroups, while for a(t) = t it reduces to the generation theorem for cosine families due to Da Prato and Giusti [63]. In general, (H2) is quite difficult to check, however there are several important special classes of operators A and kernels a(t) such that this is possible. Many of the results in subsequent sections rely on Theorem 1.3.

1. Resolvents

45

By means of Theorem 1.3 several important conclusions can be derived from existence of an exponentially bounded resolvent. Corollary 1.6 Suppose (1.1) admits a resolvent S(t) of type (M, ω). Then (i) a(λ) = 0 for Re λ > ω and Ω = {1/ a(λ) : Re λ > ω} ⊂ ρ(A); (ii) A is Abel-ergodic at 0, i.e. μ(μ − A)−1 → I strongly as |μ| → ∞, μ ∈ Ωθ = {1/ a(λ) : Re λ > ω, | arg λ| ≤ θ}, for each θ < π/2; (iii) If a(t) is real, then w.l.o.g. a(λ) > 0 for λ > ω, hence (μ0 , ∞) ⊂ ρ(A) for some μ0 > 0 and μ(μ − A)−1 → I strongly as μ → ∞. Proof: (i) has already been proved in front of Theorem 1.3. (ii) Equation (1.40) with n = 0 yields |μ(μ − A)−1 | ≤ M Re|λ| ≤ Mθ , where λ−ω −1 μ = 1/ a(λ), Re λ ≥ ω + 1, | arg λ| ≤ θ < π/2, thus μ(μ − A) is bounded for such μ. For x ∈ D(A) we have μ(μ − A)−1 x = x + (μ − A)−1 Ax → x

as |μ| → ∞,

hence μ(μ − A)−1 → I by the Banach-Steinhaus theorem. (iii) is obvious from (ii) and (i). 2

In contrary to the case of semigroups and cosine families, resolvents if they exist need not to be exponentially bounded even if the kernel a(t) involved is of class C ∞ (⺢+ ) and belongs to L1 (⺢+ ). The reason for this phenomenon is the fact that there exist kernels a ∈ C ∞ (⺢+ ) ∩ L1 (⺢+ ) with a(0) = 1 such that a ˆ(λ) admits zeros with arbitrarily large real part. By Corollary 1.6, (1.1) cannot admit an exponentially bounded resolvent. On the other hand, if A merely generates a C0 -semigroup then (1.1) for such kernel a(t) admits a resolvent by Corollary 1.4 with k(t) = a(t) ˙ and b(t) = 0. For the construction of a kernel with the described properties we refer to Desch and Pr¨ uss [91]. One should note, however, that the derivative a(t) ˙ of such kernels behaves very badly, i.e. is not Laplace transformable, ˆ˙ since otherwise a ˆ(λ) = (1 + a(λ))/λ would be nonzero for large real parts of λ. 1.6 Integral Resolvents Suppose (1.1) admits a resolvent S(t) and let a ∈ BVloc (⺢+ ). We have seen in ˙ Section 1.2 that S(t) then is differentiable and for x ∈ D(A), S(t)x is a solution of

t

˙ S(t)x = a(t)Ax + A 0

˙ )xdτ ; a(t − τ )S(τ

˙ Proposition 1.3 proved that S(t) is useful for variation of parameters formulas as well. Returning to the general case, we consider the integral equation R(t)x = a(t)x + A 0

t

a(t − τ )R(τ )xdτ,

t ≥ 0,

(1.42)

46


instead of (1.7). A solution of (1.42) is called integral resolvent for (1.1); the precise definition is as follows. Definition 1.6 A family {R(t)}t≥0 ⊂ B(X) is called integral resolvent for (1.1) if the following conditions are satisfied. (R1) R(·)x ∈ L1loc (⺢+ ; X) for each x ∈ X, and |R(t)| ≤ ϕ(t) a.e. on ⺢+ , for some ϕ ∈ L1loc (⺢+ ); (R2) R(t) commutes with A for each t ≥ 0; (R3) the integral resolvent equation holds t R(t)x = a(t)x + a(t − τ )AR(τ )xdτ for all x ∈ D(A), a.a. t ≥ 0. (1.43) 0

Suppose R(t) is an integral resolvent for (1.1), let f ∈ C(J; X) and u ∈ C(J; X) be a mild solution for (1.1). Then R ∗ u is well-defined and continuous and we obtain from (1.43) and (1.1) a ∗ u = (R − Aa ∗ R) ∗ u = R ∗ u − R ∗ Aa ∗ u = R ∗ f, i.e. R ∗ f ∈ C(J; XA ) and from (1.1) we obtain t R(t − s)f (s)ds, u(t) = f (t) + A 0

t ∈ J;

(1.44)

compare (1.44) with (1.13) above. (1.44) easily yields the uniqueness of the integral resolvent for (1.1). If both, S(t) and R(t) exist for (1.1), the relations between S and R are given by ˙ R(t)Ax = S(t)x for x ∈ D(A), t ≥ 0, (1.45) and

R(t)x = (a ∗ S)· (t)x

for x ∈ X, t ≥ 0.

(1.46)

For the case of the Cauchy problem (1.2), i.e. a(t) ≡ 1, we have R(t) ≡ S(t) while for the second order equation (1.3) S(t) = Co(t), the cosine family generated by A and R(t) = Si(t), the sine family generated by A. It is possible that R(t) exists but not S(t), and vice versa. If R(t) is an integral resolvent for (1.1) then (1.44) yields a mild solution to (1.1) in case f ∈ C(J; XA ), and even a strong solution in case f = a ∗ g with g ∈ C(J; XA ); this can be seen as in the proof of Proposition 1.3. There is a result similar to Theorem 1.3 of the last section for integral resolvents. Theorem 1.4 Let A be a closed linear unbounded operator in X with dense ∞ domain D(A) and let a ∈ L1loc (⺢+ ) satisfy 0 e−ωt |a(t)|dt < ∞. Then (1.1) admits an integral resolvent R(t) such that |R(t)| ≤ eωt ϕ(t),

for a.a. t > 0,

holds for some ϕ ∈ L1 (⺢+ ) iff the following conditions are satisfied.

(1.47)

1. Resolvents

47

(K1) a(λ) = 0 and 1/ a(λ) ∈ ρ(A) for all λ > ω; (K2) K(λ) = a(λ)(I − a(λ)A)−1 satisfies the estimates ∞

(λ − ω)n |K (n) (λ)|/n! ≤ M

for all λ > ω.

(1.48)

n=0

The proof of this result is similar to that of Theorem 1.3, based on Theorem 0.3. Since it is not needed later on, we therefore omit it here. Corollary 1.6 remains true with S(t) replaced by R(t). 1.7 Comments a) For the finite-dimensional case the solution of T = a(t)A + a ∗ AT is usually called the resolvent of (1.1). In our terminology this corresponds to the integral resolvent, i.e. we have T = RA. For equations with unbounded operators in infinite dimensions our concept of the resolvent S(t) seems to be more appropriate since it is a direct generalization of C0 -semigroups and cosine families. S(t) is sometimes called differential resolvent to distinguish it from the integral resolvent R(t). b) Corollary 1.2(i) is essentially due to Carr and Hannsgen [38], while the second part of this corollary is taken from Friedman [121]; see also Sforza [301] where mainly the second order case is treated. For L∞ -estimates of tβ μγ s(t; μ) which imply estimates of (−A)γ S(t) and S (n) (t) in this setting, see Engler [112]. c) The perturbation approach for (1.1) like in Section 1.4 was one of the first methods available to solve linear Volterra equations involving unbounded operators and has been used by many authors. It applies to much more general, even nonconvolution equations and will be taken up in Chapter II again. Therefore we postpone a discussion of the relevant literature. Perturbations of the operator A have been considered by Rhandi [291]. In this paper it is proved that in case a ∈ BVloc (⺢+ ) and B ∈ B(X) or B ∈ B(XA ) then well-posedness of (1.1) and of (1.1) with A + B instead of A are equvialent. d) Examples 1.1 and 1.2 are taken from Pr¨ uss [271]; see also Grimmer and Pr¨ uss [145] for an earlier version of Example 1.2. These papers contain another interesting example which we briefly describe now. Example 1.3 Consider the initial boundary value problem t b(t − τ )Du(τ, x)dτ, t ≥ 0, x ∈ [0, 2π], ut (t, x) = Du(t, x) − 0

u(t, 0) = ux (t, 0) = u(0, x) =

u(t, 2π),

t ≥ 0,

ux (t, 2π), t ≥ 0, u0 (x), x ∈ [0, 2π],

(1.49)

48


√ 2 where b(t) = 1/ πt and D = −2i(d/dx) + (1 − i)(d/dx). Obviously, (1.49) is of the form (1.1) with a(t) = 1 − 2 t/π and A defined in X = L2 (0, 2π) by (Au)(x) = Du(x), D(A) = {u ∈ W 2,2 (0, 2π) : u(0) = u(2π), u (0) = u (2π)}. It is not difficult to show that A does not generate a C0 -semigroup since σ(A) = {n(1 + i) + 2in2 : n ∈ ⺪} is not contained in any left halfplane. On the other hand, in Pr¨ uss [271] or Grimmer and Pr¨ uss [145] it is shown that (1.49) admits a resolvent, i.e. is well-posed. e) The generation theorem in Section 1.5 is essentially due to Da Prato and Iannelli [66] who proved it for the case a ∈ BVloc (⺢+ ). Nonscalar versions of this result will be discussed in Chapter II. f) Theorem 1.3 admits a slight extension involving only the growth bound of S(t). Theorem 1.3 : Let A be a closed linear unbounded operator in X with dense ∞ domain D(A) and let a ∈ L1loc (⺢+ ) satisfy 0 e−ωa t |a(t)|dt < ∞. Then (1.1) admits a resolvent S(t) with growth bound ω0 ∈ ⺢ iff the following conditions are satisfied. (H3) a ˆ(λ) = 0 for Re λ > max(ω0 , ωa ); if ω0 < ωa then 1/ˆ a(λ) admits analytic continuation to the halfplane Re λ > ω0 , and 1/ˆ a(λ) ∈ ρ(A) for all Re λ > ω0 ; (H4) for each ε > 0 there is a constant Mε ≥ 1 such that H(λ) = (1/ˆ a(λ) − A)−1 /(λˆ a(λ)) satisfies |H (n) (λ)| ≤ Mε n!(λ − ω0 − ε)−(n+1) ,

Re λ > ω0 + ε, n ∈ ⺞0 .

Except for the analytic continuation of 1/ˆ a(λ) in case ω0 < ωa , which will be obtained in Section 2.1, Theorem 1.3 is a simple consequence of Theorem 1.3. Observe that ω0 < 0 in Theorem 1.3 forces a(t) to be nonintegrable, i.e. ω0 < ωa . g) As in the case of the abstract Cauchy problem u˙ = Au+f which corresponds to a(t) ≡ 1, a theory of integrated resolvent and distribution resolvents even for nondensely defined operators A has been developed; see Arendt and Kellermann [13] for integrated resolvents and Da Prato and Iannelli [67] for distribution resolvents. The duality theory of resolvents has not been studied so far. i) An important practical problem is the identification problem, which can be roughly stated as follows. Suppose the space X and the operator A are given and let some (or just one) solution u(t) for different (only one) forcing functions f (t) be known; determine the kernel a(t)! This so-called ‘inverse problem’ is not considered in this book, and there are not many papers dealing with it; so far, the only ones we know of are the papers Lorenzi [221], and Lorenzi and Sinestrari [222], [223]. However it will become clear in Section 2.1 that the kernel a(t) is already uniquely determined from the knowledge of just one observation.

2

Analytic Resolvents

This section is devoted to the theory of analytic resolvents, the analog of analytic semigroups for Volterra equations of scalar type. A complete characterization of such resolvents in terms of Laplace transforms is given. In contrast to the general generation theorem of Section 1, the main result of this section, Theorem 2.1, requires conditions which are much simpler to check; this is done in several illustrating examples. The spatial regularity of analytic resolvents is studied and a characterization of analytic semigroups in these terms is derived. It is shown that analytic resolvents lead to improved perturbation results and stronger properties of the variation of parameter formulas. 2.1 Definition and First Properties In this section we consider again the Volterra equation

t

u(t) = f (t) + 0

a(t − s)Au(s)ds,

t ∈ J,

(2.1)

where a(t) and A as well as f are as in Section 1, and X is a complex Banach space. In the following, we denote by Σ(ω, θ) the open sector with vertex ω ∈ ⺢ and opening angle 2θ in the complex plane which is symmetric w.r.t. the real positive axis, i.e. Σ(ω, θ) = {λ ∈ ⺓ : | arg(λ − ω)| < θ}. We consider in this section a special class of equations of the form (2.1), namely which are such that the resolvent S(t) of (2.1) is analytic in the sense of the following definition. Definition 2.1 A resolvent S(t) for (2.1) is called analytic, if the function S(·) : ⺢+ → B(X) admits analytic extension to a sector Σ(0, θ0 ) for some 0 < θ0 ≤ π/2. An analytic resolvent S(t) is said to be of analyticity type (ω0 , θ0 ) if for each θ < θ0 and ω > ω0 there is M = M (ω, θ) such that |S(z)| ≤ M eωRez ,

z ∈ Σ(0, θ).

(2.2)

As a simple consequence of the definition of analytic resolvents we obtain estimates on the derivatives of S(t) by means of Cauchy’s integral formula. Corollary 2.1 Suppose S(t) is an analytic resolvent for (2.1) of analyticity type (ω0 , θ0 ). Then for each ω > ω0 , θ < θ0 there is M = M (ω, θ) such that |S (n) (t)| ≤ M n!eωt(1+α) (αt)−n ,

t > 0, n ∈ ⺞,

(2.3)

where α = sin θ. J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_2, © Springer Basel 1993

49

50


Proof: Let ω > ω0 and θ < θ0 be fixed and choose M = M (ω, θ) according to Definition 2.1. Cauchy’s formula then yields (n) −1 S (t) = (2πi) n! S(z)(z − t)−n−1 dz, (2.4) |z−t|=r

where r = t sin θ. Estimate (2.2) gives |S (n) (t)| ≤ ≤

(2π)−1 n!M

and this is precisely (2.3).

n!M e

π

eω(t+r cos ϕ) r−n dϕ

−π ω(t+r) −n

r

,

n ∈ ⺞, t > 0,

2

˙ Estimate (2.3) shows that for an analytic resolvent S(t) for (2.1), |S(t)| ∼ 1/t, ˙ ˙ and so |S(·)| is almost integrable at t = 0. However, unless A is bounded, |S(·)| will not be integrable near t = 0. In fact, if the latter would be the case then −ωt ˙ F (t) = S(t)e ∈ L1 (⺢+ , B(X)), hence |F(λ)| → 0 as |λ| → ∞; with F(λ) = (λ + ω)H(λ + ω) − I = (I − a(λ + ω)A)−1 − I, (I − μA)−1 is surjective for some μ ∈ ⺓, hence D(A) = X, and so A must be bounded. The case a(t) ≡ 1, i.e. the case of analytic semigroups, shows that (2.3) cannot be improved, in general. In analogy to the theory of analytic semigroups, a complete characterization of analytic resolvents of analyticity type (ω0 , θ0 ) is possible in terms of the spectrum of A and the Laplace-transform of a(t). In the sequel it will be assumed that a(t) is of exponential growth. (i) To derive these conditions assume S(t) is an analytic resolvent of type (ω0 , θ0 ) and define H(λ) = S(λ), Re λ > ω0 . Since S(z) is holomorphic in Σ(0, θ0 ), H(λ) admits analytic extension to the sector Σ(ω0 , π/2 + θ0 ), by Theorem 0.1, and for each ω > ω0 , θ < θ0 there is a constant C(ω, θ) such that the estimate |H(λ)| ≤ C(ω, θ)/|λ − ω|, λ ∈ Σ(ω, θ + π/2), (2.5) holds. (ii) Next we show that the holomorphic function a(λ) admits meromorphic extension to the same sector Σ(ω0 , θ0 + π/2). In fact, from Theorem 1.3 we know already H(λ) = (I − a(λ)A)−1 /λ, Re λ > ω1 , (2.6) ∞ −ω1 t where ω1 > ω0 is chosen large enough so that 0 |a(t)|e dt < ∞. Let x ∈ D(A) and x∗ ∈ X ∗ be such that ϕ(λ) = λ < H(λ)x, x∗ >≡< x, x∗ >; such x, x∗ always exist since otherwise λH(λ) ≡ I which implies a ˆ(λ) ≡ 0 or A = 0, in contradiction to our standing hypotheses a(t) ≡ 0 and A unbounded. ϕ(λ) is holomorphic on Σ(ω0 , θ0 + π/2), hence its derivative ϕ (λ) is so as well. For Re λ > ω1 we obtain ϕ (λ) = a (λ) < (I − a(λ)A)−2 Ax, x∗ > = a (λ)λ2 < H(λ)2 Ax, x∗ > = a (λ)ψ(λ),

2. Analytic Resolvents

51

where ψ(λ) = λ2 < H(λ)2 Ax, x∗ > is holomorphic on Σ(ω0 , θ0 + π/2), and ψ(λ) ≡ 0, since otherwise ϕ(λ) ≡ limλ→∞ ϕ(λ) =< limλ→∞ λH(λ)x, x∗ >=< x, x∗ > by Corollary 1.6, in contradiction to the choice of x and x∗ . This shows that a (λ) = ϕ(λ)/ψ(λ) is meromorphically extendible to this sector. We also have ϕ (λ)

= a (λ) < A(I − a(λ)A)−2 x, x∗ > = [ a (λ)/ a(λ)]· < λ2 H(λ)2 x − λH(λ)x, x∗ > = [ a (λ)/ a(λ)] · χ(λ),

where χ(λ) is holomorphic on Σ(ω0 , θ0 + π/2); this implies a(λ) = a (λ)χ(λ)/ϕ (λ) = χ(λ)/ψ(λ),

Re λ > ω1 ,

i.e. a(λ) admits a meromorphic continuation to Σ(ω0 , θ0 + π/2). (iii) Relation (2.6) can now be extended to Σ(ω0 , θ0 + π/2) as follows. Let Ω = {λ ∈ Σ(ω0 , θ0 + π/2) : λ = 0, a ˆ(λ) = 0, ∞}; for x ∈ D(A) we have λH(λ)(I − a(λ)A)x = x

for all λ ∈ Ω,

(2.7)

since this relation holds for Re λ > ω1 , i.e. I − a(λ)A is injective for all λ ∈ Ω. On the other hand, for Re λ > ω1 , H(λ) commutes with A, i.e. for each μ ∈ ρ(A) we have (μ − A)−1 H(λ) = H(λ)(μ − A)−1 . By analytic continuation, this relation holds on all of Ω and therefore H(λ)D(A) ⊂ D(A) for all λ ∈ Ω, and AH(λ)x is analytic on Ω. But this implies also the identity λ(I − a(λ)A)H(λ)x = x,

λ ∈ Ω,

(2.8)

for all x ∈ D(A), since this is valid for Re λ > ω1 . Therefore, the operators λ(I − a(λ)A) have dense ranges in X, and so they are invertible for all λ ∈ Ω, and (2.6) holds on Ω as well. (iv) If λ0 ∈ Σ(ω0 , θ0 + π/2) is such that a(λ0 ) = 0, then - since λ0 is isolated we may choose a small circle Γ around λ0 contained in Σ(ω0 , θ0 + π/2), and apply the Cauchy integral formula to the operator-valued function λH(λ) to obtain λH(λ) λH(λ) − I A = Aλ0 H(λ0 ) = A(2πi)−1 dλ = (2πi)−1 dλ, a(λ) Γ λ − λ0 Γ (λ − λ0 ) i.e. A is bounded, in contradiction to the standing hypotheses on A. Thus a(λ) does not have zeros in Σ(ω0 , θ0 + π/2). (v) If λ0 ∈ Σ(ω0 , θ0 + π/2) is a pole of a(λ), say of order n, then the Laurentexpansion a(λ) = a−n (λ − λ0 )−n + (λ − λ0 )−n+1 ϕ(λ) with a holomorphic function ϕ(λ) near λ0 and Taylor-expansion of λH(λ) show that λH(λ) has a zero of order n; passing to the limit λ → λ0 in (2.7) and (2.8) we obtain with Hn = lim λH(λ)(λ − λ0 )−n λ→λ0

52


the identity −a−n Hn Ax = −a−n AHn x = x

for all x ∈ D(A).

1 Since −a−n = 0 this yields 0 ∈ ρ(A) and Hn = − a−n A−1 . Thus a(λ) may only have poles in Σ(ω0 , θ0 + π/2) if 0 ∈ ρ(A); note that in case ω0 < 0 holds, λ = 0 must be a pole for a(λ).

2.2 Generation of Analytic Resolvents It turns out that the conditions derived so far are characterizing analytic resolvents of type (ω0 , θ0 ). Theorem 2.1 Let A be a closed linear unbounded operator in X with dense ∞ domain D(A) and let a ∈ L1loc (⺢+ ) satisfy 0 |a(t)|e−ωa t dt < ∞ for some ωa ∈ ⺢. Then (2.1) admits an analytic resolvent S(t) of analyticity type (ω0 , θ0 ) iff the following conditions hold. (A1) a(λ) admits meromorphic extension to Σ(ω0 , θ0 + π/2); (A2) a(λ) = 0, and 1/ a(λ) ∈ ρ(A) for all λ ∈ Σ(ω0 , θ0 + π/2); (A3) For each ω > ω0 and θ < θ0 there is a constant C = C(ω, θ) such that H(λ) = (1/ˆ a(λ) − A)−1 /(λˆ a(λ)) satisfies the estimate |H(λ)| ≤ C/|λ − ω|

for all λ ∈ Σ(ω, θ + π/2).

(2.9)

Proof: It remains to prove the sufficiency part. Let ω > ω0 and θ < θ0 be given and choose θ ∈ (θ, θ0 ) as well as C = C(ω, θ ) from (A3). Define S(t) by S(z) = (2πi)−1 eλz H(λ)dλ, z ∈ Σ(0, θ), (2.10) ΓR

where ΓR denotes the contour consisting of the two rays ω + ireiθ and ω − ire−iθ with r ≥ R and the larger part of the circle |λ − ω| = R connecting these rays. Let z = teiϕ , R = 1/t, and α = sin(θ − θ); estimating (2.10) we obtain |S(z)| ≤ (C/2π) eRe(λz) |λ − ω|−1 |dλ| ΓR ∞ π ωRez −αr ≤ (C/π)e { e dr/r + ecos ϕ dϕ} 1

0

≤ M eωRez . This estimate shows that the integral in (2.10) is absolutely convergent for z ∈ Σ(0, θ), hence S(z) is holomorphic in this region and (2.2) of Definition 2.1 is satisfied. Moreover, for the Laplace-transform of S(t) we obtain for λ > ω ∞ ∞ S(λ) = e−λt S(t)dt = (2πi)−1 e−λt eμt H(μ)dμdt 0

0

Γ0

2. Analytic Resolvents −1

53 (

= (2πi)

Γ0

∞

e

−(λ−μ)t

−1

dt)H(μ)dμ = (2πi)

0

Γ0

H(μ)(λ − μ)−1 dμ

= H(λ), where we used Fubini’s theorem and Cauchy’s integral formula. Thus H(λ) is the Laplace-transform of S(t) and therefore by Theorem 1.3, S(t) is a resolvent for (2.2). Finally, since ω > ω0 and θ < θ0 are arbitrary, S(t) is an analytic resolvent of type (ω0 , θ0 ). 2

For an analytic resolvent S(t) for (2.1) we so far only know that S(z) → I strongly as z → 0+ for real z, and so a natural question is whether this convergence is true in every sector Σ(0, θ) with θ < θ0 . The answer is in the affirmative. Corollary 2.2 Suppose S(t) is an analytic resolvent for (2.1) of type (ω0 , θ0 ). Then S(z)x → x for z ∈ Σ(0, θ) as z → 0, for each x ∈ X and θ < θ0 . Proof: Let x ∈ X and x∗ ∈ X ∗ and consider ϕ(z) = < S(z)x, x∗ >. Since ϕ(t) → ϕ(0) = < x, x∗ > as t → 0+ and ϕ is holomorphic and bounded on Σ(0, θ0 ), a well-known result about the boundary behaviour of analytic functions (cp. Duren [106], Theorem 1.3) implies ϕ(z) → ϕ(0) as z → 0+, uniformly for z ∈ Σ(0, θ), for all θ < θ0 . Therefore, the Laplace-transform ϕ(λ) has the convergence property λϕ(λ) = < λH(λ)x, x∗ > → < x, x∗ >= ϕ(0)

as |λ| → ∞,

where λ ∈ Σ(ω, θ + π/2), for any ω > ω0 , θ < θ0 ; similarly, φ(λ) = < λ2 H(λ)2 x, x∗ > → < x, x∗ > as |λ| → ∞, λ ∈ Σ(ω, θ + π/2), where x and x∗ are arbitrary. Therefore a(λ) = χ(λ)/ψ(λ) → 0

as |λ| → ∞,

uniformly in λ ∈ Σ(ω, θ + π/2),

for each ω > ω0 , θ < θ0 , where χ(λ) and ψ(λ) are defined as in (ii) of Section 2.1, i.e. ψ(λ) = λ2 < H(λ)2 Ax, x∗ >,

χ(λ) = < λH(λ)(λH(λ)x − x), x∗ >,

with x ∈ D(A) and x∗ ∈ X ∗ such that < Ax, x∗ > = 0. (2.10) and (2.7) then imply for x ∈ D(A) −1 S(z)x − x = (2πi) eλz H(λ) a(λ)Axdλ, ΓR

hence with z = teiϕ , R = 1/t, as in the proof of Theorem 2.1 |S(z)x − x| ≤ M eωRez |Ax| sup{| a(λ)| : |λ| ≥ 1/t, λ ∈ Σ(ω, θ + π/2)} → 0

54


as |z| = t → 0. Thus S(z)x → x as z → 0, z ∈ Σ(0, θ) for each x ∈ D(A), and since S(z) is uniformly bounded for such z, by the Banach-Steinhaus theorem S(z) → I strongly as z → 0, z ∈ Σ(0, θ). 2

Another natural question which arises in connection with analytic resolvent is what regularity is imposed on a(t) by conditions (A1) ∼ (A3). It turns out that a(t) itself admits analytic continuation to the same sector Σ(0, θ0 ) of analyticity of the resolvent S(t). Corollary 2.3 Suppose S(t) is an analytic resolvent for (2.1) of analyticity type (ω0 , θ0 ). Then a(t) admits analytic extension to Σ(0, θ0 ). Furthermore, on each subsector Σ(0, θ), θ < θ0 , there is a decomposition of the form a(z) = pj (z)eλj z + a1 (z), z ∈ Σ(0, θ), (2.11) j

where the λj denote the finitely many poles of a(λ) contained in Σ(ω, θ + π/2), the pj (z) are polynomials, and a1 (z) is analytic in Σ(0, θ) and satisfies |a1 (z)| ≤ CeωRez /|z|, za1 (z) → 0

z ∈ Σ(0, θ),

(2.12)

as z → 0, z ∈ Σ(0, θ0 ),

(2.13)

and a(λ) may only have poles in case 0 ∈ ρ(A). Proof: Let ω > ω0 and θ < θ0 be fixed but arbitrary otherwise, and choose ω ∈ (ω0 , ω), θ ∈ (θ, θ0 ) such that no poles of a(λ) are located on the rays ω ± ire±iθ . From the proof of Corollary 2.2 we already know a(λ) → 0 as |λ| → ∞, λ ∈ Σ(ω , θ + π/2), hence there are only finitely many poles λ1 , . . . , λn ∈ Σ(ω , θ + π/2) of a(λ). Deforming the contour Γ in the complex inversion formula for the Laplace-transform of a(t) into Γ0 , by the theorem of residues we obtain a(z)

=

n

−1

Res(e a(λ))λ=λj + (2πi) λz

j=1

=

n

Γ0

eλz a(λ)dλ

pj (z)eλj z + a1 (z),

j=1

i.e. decomposition (2.11) holds. a1 (z) is estimated as follows ∞ |a1 (z)| ≤ sup{| a(λ)| : λ ∈ Γ0 }π −1 eRe(λz) |dλ| 0

≤

M eω Rez /|z cos(π/2 + θ − θ)| ≤ CeωRez /|z|,

z ∈ Σ(ω, θ + π/2),

i.e. (2.12) holds. (2.13) follows similarly by choosing ΓR as the integration contour, and the last statement is already clear from (v) in Section 2.1. 2


55

2.3 Examples It is worthwhile to mention that Hille’s generation theorem for analytic C0 -semigroups cp. Hille-Phillips [180], Theorem 12.8.1, is a special case of Theorem 2.1. In fact, this corresponds to the case a(t) ≡ 1; (A1) is then trivially satisfied since a(λ) = 1/λ and (A2) and (A3) reduce to Σ(ω0 , θ0 + π/2) ⊂ ρ(A)

and

|(λ − A)−1 | ≤ C/|λ − ω|,

λ ∈ Σ(ω, θ + π/2),

which are precisely Hille’s conditions. Typical examples for kernels a(t) and operators A satisfying (A1) ∼ (A3) are given in Example 2.1 Consider the kernels a(t) = tβ−1 /Γ(β),

t>0

where β ∈ (0, 2) and Γ denotes the gamma-function. Let us examine for which operators (2.1) admits a bounded analytic resolvent S(t). For the Laplace-transform of a(t) we obtain a(λ) = λ−β , Re λ > 0, hence a(λ) admits analytic extension to the complex plane sliced along the negative real axis; thus (A1) holds for Σ(0, π). The function 1/ a(λ) maps the sector Σ(0, θ0 +π/2) onto the sector Σ(0, θ1 ), θ1 = β(θ0 +π/2), hence (A2) with ω0 = 0 is equivalent to ρ(A) ⊃ Σ(0, θ1 ), for some θ1 > βπ/2. Finally, S(t) will be bounded in some sector Σ(0, θ) if (A3) holds uniformly in ω > 0, which is equivalent to |μ(μ − A)−1 | ≤ M

for all μ ∈ Σ(0, βπ/2).

Thus the pair (tβ−1 /Γ(β), A) generates a bounded analytic resolvent iff ρ(A) ⊃ Σ(0, βπ/2)

and

|μ(μ − A)−1 | ≤ M

for all μ ∈ Σ(0, βπ/2).

Example 2.2 An important class of kernels a(t) which satisfy (A1) is the class of completely monotonic kernels, i.e. kernels which are represented as Laplace transforms of positive measures ∞ a(t) = e−st dα(s), t > 0, 0

∞ with α(t) nondecreasing, and such that 1 dα(s)/s < ∞ (the latter is equivalent to a ∈ L1loc (⺢+ )). a(λ) is then given by the integral representation ∞ 1 a(λ) = dα(s), Re λ > 0, λ+s 0 hence a(λ) admits analytic extension to Σ(0, π). The kernels of Example 1 with β ≤ 1 are of this kind. Moreover, the decomposition ∞ ∞ σ+s 1 a(σ + iρ) = dα(s) − iρ dα(s) 2 + ρ2 2 + ρ2 (σ + s) (σ + s) 0 0

56


shows a(λ) = 0 on ⺓ \ ⺢− and | arg a(λ)| ≤ | arg λ|

for all λ ∈ Σ(0, π).

Thus if A generates an analytic C0 -semigroup bounded in some sector Σ(0, θ), then (A2) and (A3) are satisfied since then |μ(μ − A)−1 | ≤ M,

for all μ ∈ Σ(0, θ + π/2).

Therefore by Theorem 2.1, (2.1) admits an analytic resolvent S(t) of type (0, θ). Let us summarize this in Corollary 2.4 Let a ∈ C(0, ∞) ∩ L1 (0, 1) be completely monotonic and let A generate an analytic semigroup T (t) such that |T (t)| ≤ M on Σ(0, θ). Then (2.1) admits an analytic resolvent S(t) of type (0, θ). Example 2.3 Consider the kernel ∞ a(t) = tρ−1 /Γ(ρ)dρ,

t > 0;

0

its Laplace transform is given by a(λ) = 1/ log λ. The function log λ maps the sector Σ(0, θ) onto the strip Sθ = {z ∈ ⺓ : |Im z| < θ}. Hence if A is such that S π/2 ⊂ ρ(A) and |(μ − A)−1 | ≤ C/|μ| holds on S π/2 then |H(λ)| ≤ C/|λ| for λ ∈ Σ(0, π/2 + θ), θ sufficiently small and Theorem 2.1 yields a bounded analytic resolvent S(t). On the other hand, if A is such that Σ(0, φ) ⊂ ρ(A) for some φ > 0 and |(μ−A)−1 | ≤ C/|μ|, then 1/ a(λ) maps Σ(ω, π/2+θ) into the halfstrip Sπ/2+θ ∩{μ ∈ ⺓ : Re μ ≥ R} where R = log(ω cos θ). Choosing ω large enough this halfstrip will be contained in Σ(0, φ), hence Theorem 2.1 yields a resolvent of type (ω, θ), where ω ≥ exp[(π/2 + θ)/ sin φ]/ cos θ. This example shows that there are kernels such that (2.1) admits an analytic resolvent for every operator which is such that (α, ∞) ⊂ ρ(A) for some α ≥ 0 and |μ(μ − A)−1 | ≤ C as μ → ∞, in particular for every sectorial operator, cp. Section 8. Note that the singularity of a(t) at t = 0 is quite strong, although a(t) is still integrable at t = 0; in fact, a(t) ∼ 1/(t ln2 t) as t → 0 can be shown. 2.4 Spatial Regularity For the special case a(t) ≡ 1, an analytic resolvent for (2.1) becomes the analytic semigroup S(t) = eAt generated by A. The semigroup property then yields the relations S (n) (t) = An S(t), t > 0, n ∈ ⺞, ∞ and therefore we have R(S(t)) ⊂ n=1 D(An ) for t > 0, as well as the estimates |An S(t)| = |S (n) (t)| ≤ M n!eωt(1+α) (αt)−n ,

t > 0, n ∈ ⺞,


57

by Corollary 2.1. For the general case, we still obtain R(S(t)) ⊂ D(A), but nothing more, in general, as the next theorem shows. Theorem 2.2 Suppose S(t) is an analytic resolvent for (2.1) of type (ω0 , θ0 ). Then (i) R(S(t)) ⊂ D(A) for all t > 0, and for each ω > ω0 , θ < θ0 , there are π constants c, C > 0 and β ∈ ( 2θπ0 , 2θ ) such that β

|AS(t)| ≤ Ceωt (tβ + ec/t ),

t > 0.

(2.14)

(ii) If in addition, there are ω > ω0 , 0 < θ < θ0 , c > 0, and α > 0 such that | a(λ)| ≥ c(|λ − ω|α + 1)−1 for all λ ∈ Σ(ω, π/2 + θ),

(2.15)

then there is a constant C > 0 such that |AS(t)| ≤ Ceωt (1 + t−α ),

t > 0.

(2.16)

(iii) If (2.15) holds and if there is a set E ⊂ ⺢+ which is not discrete and such that A2 S(t) is bounded for each t ∈ E, then a(t) ≡ a0 , and a0 A generates an analytic semigroup T (t). The proof of (i) is based on the following Lemma on holomorphic functions on the unit disc ⺔ = {z ∈ ⺓ : |z| < 1}; although the proof is standard it is included here for the sake of completeness. Lemma 2.1 Suppose φ1 (z), φ2 (z) are bounded holomorphic functions on the open unit disk ⺔ ⊂ ⺓ such that φ2 (z) = 0 on ⺔. Then there is a constant c > 0 such that ϕ(z) = φ1 (z)/φ2 (z) satisfies |ϕ(z)| ≤ exp(c ·

1 + |z| ) for all z ∈ ⺔. 1 − |z|

(2.17)

Proof: By the canonial factorization theorem for bounded analytic functions (cp. Duren [106], Theorem 2.8), there are real numbers γj , functions μj (t) ≥ 0 with log μj (·) ∈ L1 (0, 2π), singular nonnegative measures νj on [0, 2π], and Blaschkeproducts bj (z) such that φj (z) = eiγj · bj (z) · exp(

1 2π

2π 0

2π it eit + z e +z log μ (t)dt) · exp( dνj (t)) j eit − z eit − z 0

holds for j = 1, 2. Since φ2 (z) = 0 on ⺔ we have b2 (z) ≡ 1; on the other hand, |b1 (z)| ≤ 1 and with z = reiϕ 0 ≤ Re

eit + z 1 − r2 1 − r2 1+r 1 + |z| = ≤ = = . it 2 e −z 1 − 2r cos(ϕ − t) + r (1 − r)2 1−r 1 − |z|

58


From this we obtain (2.17) with 2π c = Var ν1 |2π 0 + Var ν2 |0 + | log μ1 |1 + | log μ2 |1 . 2

Proof of Theorem 2.2: (i) By means of Lemma 2.1 we first derive an estimate for 1/ a(λ) on a sector Σ(ω, π2 + θ), where ω > ω0 , θ < θ0 , are fixed but arbitrary otherwise. Choose β = 2θπ , with θ < θ < θ0 , ω > ω > ω0 , and consider the function 1+z p λ(z) = ω + ( ) , z ∈ ⺔, (2.18) 1−z where p = 1+1/β. λ maps the unit disk ⺔ conformally onto the sector Σ(ω , π2 +θ ). a(λ) has only finitely many poles within this sector, since a(λ) is bounded as |λ| → ∞, λ ∈ Σ(ω , π2 + θ ), and therefore a(λ) can be factored according to (λ − ω0 )m a(λ) = n · χ2 (λ) = χ2 (λ)/χ1 (λ), pj j=1 (λ − λj ) where the λj denote the poles of a(λ) within Σ(ω , π2 +θ ), pj their orders, and m =

j pj . Obviously, χ1 (λ) and χ2 (λ) are both holomorphic on Σ(ω , π/2 + θ ), χ1 (λ) is bounded and has the zeros λj , whereas χ2 (λ) is nonvanishing and bounded, too. Let ψj (z) = χj (λ(z)); then Lemma 2.1 yields the estimate |

1 ψ1 (z) 1 + |z| | = |ϕ(z)| = | | ≤ exp(c · ) a(λ) ψ2 (z) 1 − |z|

for some c > 0. Defining μ = other hand

1+z 1−z

on ⺔,

we obtain from (2.18) μ = (λ − ω )1/p ; on the

1 + |z| |μ + 1| + |μ − 1| (|μ + 1| + |μ − 1|)2 (1 + |μ|)2 = = ≤ , 1 − |z| |μ + 1| − |μ − 1| |μ + 1|2 − |μ − 1|2 Re μ and so with α =

1 p

|μ| = |λ − ω |α , Re μ = Re(λ − ω )α = |λ − ω |α cos(arg(λ − ω )α). Since for λ ∈ Σ(ω , π2 + θ) | arg(λ − ω )|α ≤ (

π π π + θ)/p < (1 + 1/β)/(1 + 1/β) = , 2 2 2

these estimates yield |

1 | ≤ a(λ) ≤

exp(c ·

(1 + |λ − ω |α )2 ) |λ − ω |α

exp(c(|λ − ω|α + 1)),

λ ∈ Σ(ω,

π + θ). 2

(2.19)


59 −1

Next, we use representation (2.10) with R = t−(1−α) to show (2.14). In fact AS(t) = (2πi)−1 eλt AH(λ)dλ Γ R 1 −1 = (2πi) eλt [H(λ) − 1/λ] dλ, (2.20) a(λ) ΓR hence

|AS(t)|

≤ C

eReλt ec|λ−ω|

α

ΓR

≤ Ceωt

∞ R β

e−rt sin θ er

≤ Ce (t + e ωt

ctβ

|dλ| |λ − ω| α

c dr

r

π 2 +θ

+ Ceωt

eRt cos ϕ ecR dϕ α

0

);

in particular R(S(t)) ⊂ D(A) for each t > 0 and (2.14) holds. (ii) If (2.15) holds, then we obtain from (2.20) with R = 1/t |dλ| |AS(t)| ≤ C eReλt (1 + |λ − ω|α ) |λ − ω| ΓR ∞ π/2+θ dr ≤ Ceωt e−rt sin θ (1 + rα ) + Ceωt eRt cos ϕ (1 + Rα )dϕ r R 0 ≤ Ceωt (1 + t−α ), t > 0, and so (2.16) follows. (iii) Suppose A2 S(t) is bounded for t ∈ E, where E ⊂ ⺢+ is nondiscrete. Define the function c(z) by means of 1 c(z) = eλz [λ a(λ)]−1 dλ, | arg z| ≤ θ < θ0 ; 2πi Γ0 c(z) is holomorphic in the sector Σ(0, θ) and from (2.20) we obtain c(t) = 0 for each t ∈ E, since A is unbounded by assumption. Since E is nondiscrete this implies c(z) ≡ 0, by uniqueness of holomorphic functions. Similarly, for some n > α − 2, we define b(z) by means of 1 b(z) = eλz [λn+2 a(λ)]−1 dλ, | arg z| ≤ θ < θ0 ; 2πi Γ0 b(z) is holomorphic on Σ(0, θ), and we have b(n+1) (z) = c(z) = 0 on this sector, i.e. b(z) is a polynomial of order at most n. By the choice of n, the Laplace transform of b(t) becomes b(λ) = [λ(n+2) a(λ)]−1 , λ > ω; use Fubini’s theorem and the Cauchy integral formula. On the other hand, since b(t) is a polynomial of order at most n, we have b(λ) =

n 0

bn−j λ−j−1 ,

60


for some constants bk ∈ ⺓. Therefore, we obtain the representation 1 = bj λj+1 , a(λ) 0 n

i.e. the function 1/ a(λ) is a polynomial, too. If the degree of this polynomial is c greater than one, then it maps the sets Σ(ω, π/2 + θ) ∩ BR (0) for large R onto sets containing the complement of a disk. By Theorem 2.1 this implies that ρ(A) contains the complement of a disk, and |μ(μ − A)−1 | ≤ C as |μ| → ∞; thus A is bounded, thereby contradicting our standing assumption. Therefore 1/ˆ a(λ) = b0 λ, or in other words a(t) ≡ a0 . But then S(t) = ea0 At , i.e. a0 A generates an analytic semigroup. The proof is complete. 2

2.5 Perturbed Equations Suppose that (2.1) admits an analytic resolvent S0 (t) and consider the perturbed equation u(t) = f (t) + (a + a ∗ dk) ∗ Au(t) + b ∗ u(t), t ∈ J, (2.21) 0 where a, A, f are as before, b ∈ L1loc (⺢+ ) and k ∈ BVloc (⺢+ ). In contrast to Theorem 1.2, no further assumptions are needed to obtain a resolvent for (2.2).

Theorem 2.3 Suppose (2.1) admits an analytic resolvent S0 (t) of analyticity type 0 (ω0 , θ0 ) and let b ∈ L1loc (⺢+ ), k ∈ BVloc (⺢+ ). Then (2.21) admits an exponentially bounded resolvent S(t). Proof: Without loss of generality we may assume that b and dk are Laplace transformable, i.e.

∞ 0

|dk(t)|e−ωt < ∞

∞

and 0

|b(t)|e−ωt < ∞;

even more, since k(t) does not have a jump at zero, choosing ω large enough we can achieve ∞ ∞ −ωt |b(t)|e + |dk(t)|e−ωt ≤ η, (2.22) 0

0

where η > 0 is any prescribed small number. The resolvent S(t) for (2.21) is the solution of S(t) = I + Aa ∗ S(t) + dk ∗ Aa ∗ S(t) + b ∗ S(t), and so its Laplace-transform is given by 1 S(λ) = (1 − b(λ) − (1 + dk(λ)) a(λ)A)−1 . λ

t ∈ ⺢+ ,

(2.23)


61

Denote the resolvent kernel of b(t) by r(t), i.e. r(t) is the solution of r = b + b ∗ r, and let l(t) be defined by l = k + 1 ∗ r + k ∗ r. Rewrite S(λ) as follows. 1 a(λ)A)−1 S(λ) = (1 + r(λ)) (1 − a(λ)A − dl(λ) λ 1 a(λ)A(1 − a(λ)A)−1 = (1 + r(λ))(1 − dl(λ) a(λ)A)−1 )−1 (1 − λ ∞ 1 n = (1 + r(λ)) [dl(λ)] [ a(λ)A(1 − a(λ)A)−1 ]n (1 − a(λ)A)−1 λ n=0 = (1 + r(λ))

∞

n [dl(λ)] Hn (λ),

λ > ω,

(2.24)

n=0

where

a(λ)n n A (1 − a(λ)A)−(n+1) , λ > ω, n ∈ ⺞0 . (2.25) λ We now recover the operator-valued functions Hn (λ) as Laplace transforms of functions Sn (t). For this purpose note that from Theorem 2.1 we obtain constants ω1 > max(0, ω0 ) and θ ∈ (0, θ0 ) such that Hn (λ) =

|λ| , |λ − ω1 |

|(1 − a(λ)A)−1 | ≤ C ·

λ ∈ Σ(ω1 , θ)

holds; for λ ∈ Σ(2ω1 , θ) we therefore have, with a probably different constant C, |H0 (λ)| ≤

C , |λ − 2ω1 |

|ˆ a(λ)A(1 − a(λ)A)−1 | ≤ C.

Let ω > 2ω1 be sufficiently large, and consider the integrals Sn (t) = (2πi)−1 eλt Hn (λ)dλ, t > 0,

(2.26)

ΓR

where ΓR denotes again the contour as in Section 2.2. It is easily seen that Sn (t) admits analytic extension to a sector Σ(0, θ1 ) and that |Sn (t)| ≤ C n · M eωt ,

t > 0, n ∈ ⺞0 ,

(2.27)

is satisfied. Finally, from (2.24) we obtain the resolvent S(t) for (2.21). S(t) = (δ0 + r) ∗

∞

(dl)∗n ∗ Sn (t),

t > 0,

(2.28)

n=0

and the series converges absolutely and uniformly on bounded intervals. In fact, from (2.22) and by the definition of r and l, we obtain ∞ |dl(s)|e−ωs )n |dl∗n ∗ Sn (t)| ≤ M eωt · C n ( 0

≤ M eωt (2ηC/(1 − η))n ≤ M 2−n eωt ,

62


provided η is chosen small enough, i.e. ω is large enough. Since Sn (t) is even uniformly continuous for t > 0 and commutes with A it follows from uniqueness of the Laplace-transform that S(t) defined by (2.28) is the resolvent for (2.21). 2

Note that in the contrast to the functions Sn (t), the resolvent S(t) of (2.21) is not analytic, in general, it even need not be continuous in B(X), as Example 1.1 shows. 2.6 Maximal Regularity Suppose S(t) is an analytic resolvent for (2.1), and let f ∈ C(J; X), where J = [0, T ]. Since S(t) belongs to C 1 ((0, ∞); B(X)), a much better behaviour of the variation of parameters formula (1.8) can be expected, which we now employ in the form t ˙ − s)f (s)ds, t ∈ J. u(t) = f (t) + S(t (2.29) 0

For (2.29) to make sense, continuity of f will in general not be enough, however, if f is Hölder-continuous with exponent α ∈ (0, 1), i.e. f ∈ C α (J; X), then u(t) given by (2.29) is well-defined and a mild solution of (2.1). We denote by C0α (J; X) the spaces C0α (J; X) = {f : J → X : f (0) = 0, |f |α < ∞} normed by

|f |α = sup{|f (t) − f (s)|(t − s)−α : 0 ≤ s < t ≤ T };

obviously C0α (J; X) is a Banach space. It turns out that the solution operator f → u is leaving these spaces invariant, (2.1) has the property of maximal regularity of type C α . Theorem 2.4 Suppose (2.1) admits an analytic resolvent S(t), let f ∈ C(J; X), and u(t) be defined by (2.29), where α ∈ (0, 1). Then (i) if f ∈ C0α (J; X) then u ∈ C0α (J; X) is a mild solution of (2.1); (ii) if f ∈ C0α (J; XA ) then u ∈ C0α (J; XA ) is a strong solution of (2.1); (iii) if f = a ∗ g, g ∈ C0α (J; X), then u ∈ C0α (J; XA ) is a strong solution of (2.1). Proof: (a) Let f ∈ C0α (J; X); rewrite (2.29) as t u(t) = S(t)f (t) + S (t − s)(f (s) − f (t))ds, 0

t ∈ J,

and estimate according to |u(t)|

≤ |S(t)||f (t)| + ≤ tα MT |f |α +

t 0 t

0

|S (t − s)||f (s) − f (t)|ds

MT (t − s)−1 · |f |α (t − s)α ds

≤ tα MT |f |α + MT |f |α · tα /α,

(2.30)


63

where MT = max{|S(t)|, t|S (t)| : t ∈ J}. This estimate shows that u(t) defined by (2.29), or equivalently by (2.30) exists, is bounded on J, and satisfies u(0) = 0. (b) To see that u(t) is again H¨ older-continuous let t, t ∈ J, h = t − t > 0, and decompose as follows. u(t) − u(t) = (S(t) − S(t))f (t) + S(h)(f (t) − f (t)) t + S (t − τ )(f (τ ) − f (t))dτ

t t

+ 0

(S (h + τ ) − S (τ ))(f (t − τ ) − f (t))dτ

= I1 + I2 + I3 + I4 . The integrals Ij are estimated as follows. |I1 | ≤ |S(t) − S(t)||f (t)| ≤ MT log(t/t) · |f |α · t

α

≤ MT |f |α hα /α, where we used the elementary inequality log(1 + ρ) ≤ ρα /α

for all ρ > 0, α ∈ (0, 1),

and f (0) = 0. |I2 | ≤ |S(h)||f (t) − f (t)| ≤ MT |f |α hα , since f ∈ C0α (J; X).

t

|I3 | ≤ MT |f |α

(t − τ )α−1 dτ = MT |f |α hα /α.

t

To estimate I4 we observe that from Corollary 2.1 we have |S

(t)| ≤ MT t−2 ,

t ∈ J,

where MT is large enough, hence

t+h

|S (t + h) − S (t)| ≤ t

|S

(τ )|dτ ≤ MT ·

h t(t + h)

for all t, t + h ∈ J. This yields

|I4 |

t/h hτ α−1 (τ + h)−1 dτ = MT |f |α hα τ α−1 (τ + 1)−1 dτ 0 0 ∞ α α−1 −1 α ≤ MT |f |α h τ (τ + 1) dτ = MT |f |α h π/ sin(απ). ≤ MT |f |α

t

0

Thus u ∈ C0α (J; X) and for some constant depending only on J we have the estimate |u|α ≤ C|f |α / sin(απ). (2.31)

64


(c) We next show that u(t) is a mild solution of (2.1); this follows from a simple ap∞ proximation ∞argument. Let ϕε ∈ C0 (0, ∞) be such that supp ϕε ⊂ (0, ε), ϕε (t) ≥ 0 on ⺢+ , 0 ϕε (t)dt = 1, and define fε = ϕε ∗ f. Then fε ∈ C 1 (J; X) and so uε = ϕε ∗ u is the mild solution of (2.1) with f replaced by fε . Since fε → f and uε → u, hence also a ∗ uε → a ∗ u in C(J; X) we obtain Aa ∗ uε = uε − fε → u − f, and from closedness of A there follows a ∗ u ∈ C(J; XA ) and Aa ∗ u = u − f , i.e. u is a mild solution (2.1). (d) Part (ii) follows directly from (i) since A commutes with the resolvent S(t). (e) Let v ∈ C0α (J; X) defined by (2.29) with f replaced by g denote the mild solution of v = g + Aa ∗ v; (2.32) and let u be defined by (2.29); then u = a∗ v belongs to C α (J; XA ) and convolving (2.32) with a we see that u is a strong solution of (2.1). 2

Corollary 2.5 Under the assumptions of Theorem 2.4, there is a constant C > 0 depending only on T , A and a(t) such that u(t) defined by (2.29) satisfies the following estimates. (i) |u|α ≤ C|f |α / sin(πα), f ∈ C0α (J; X); (ii) |Au|α ≤ C|Af |α / sin(πα), f ∈ C0α (J; XA ); (iii) |Au|α ≤ C|g|α / sin(πα), f = a ∗ g, g ∈ C0α (J; X). Note that the case f (0) = x = 0 can be reduced to f (0) = 0; rewrite (2.29) as u(t) = (f (t) − x) +

0

t

S (t − τ )(f (τ ) − x)dτ + S(t)x.

(2.33)

Thus if f ∈ C α (J; X) with f (0) = x = 0 it is not always true that u ∈ C α (J; X) holds. However, u(t) given by (2.33) still is a mild solution of (2.1) for each x ∈ X. From Theorem 2.4 and the properties of S(t) we have Corollary 2.6 Let S(t) be an analytic resolvent for (2.1), let u(t) be defined by (2.33) and α ∈ (0, 1). Then (i) if f ∈ C α (J; X) then u is a mild solution of (2.1); (ii) if f ∈ C α (J; XA ) then u is a strong solution of (2.1); (iii) if f = x + a ∗ g with x ∈ D(A) and g ∈ C α (J; X) then u is a strong solution of (2.1). In Section 3.5 it will be shown that Theorem 2.4 and Corollaries 2.5, 2.6 remain valid for the perturbed equation (2.21). This is at first sight somewhat surprising since the resolvent of (2.21) does not satisfy estimates (2.3) for n = 1, 2 which were used in the proof of Theorem 2.4. 2.7 Comments a) Analytic resolvents have been introduced by Da Prato and Iannelli [65] who


65

also proved a weaker version of the sufficiency part of Theorem 2.1; see also Grimmer and Pritchard [144], and Da Prato and Iannelli [68]. The neccessity part of Theorem 2.1 is new as is Corollary 2.3 showing the restriction on the kernel a(t). Examples 2.1, 2.2, and 2.3 are standard, although Example 2.3 is probably not so well-known. b) The second part of Theorem 2.2 in a somewhat weaker form is due to Da Prato, Iannelli and Sinestrari [70]; parts (i) and (iii) of Theorem 2.2 seem to be new. A counterexample of Desch and Pr¨ uss [91] shows that, in the setting of Theorem 2.2, in general (2.15) does not hold, and that the estimate (2.19) leading to (2.14) is optimal. On the other hand, if σ(A) contains a halfray [r, ∞)eiϕ or maybe arg a ˆ(λ) is bounded by some θ ∈ (0, ∞) then (2.15) is valid, as Proposition 3.2 and Lemma 8.1 show. c) The perturbation result in Section 2.5 seems to be new; a much more general version is presented in Chapter II. It is also not difficult to show that perturbations A+B of the involved operator A are possible as in the case of analytic semigroups. If B satisfies D(B) ⊃ D(A), and for each ε > 0 there is C(ε) > 0 such that |Bx| ≤ ε|Ax| + C(ε)|x|,

x ∈ D(A),

holds, then the (a, A + B) generates an analytic resolvent whenever (a, A) does so, and the angle θ0 is the same, only the growth bound ω0 possibly changes. This follows from simple estimates in frequency domain, which employ Theorem 2.1 and a ˆ(λ) → 0 as |λ| → ∞, in Σ(ω, θ), θ < π/2 + θ0 , ω > ω0 . d) It is not clear in general whether an analytic resolvent S(t) for (2.1) is differentiable in the sense of Definition 1.4, i.e. whether the integral resolvent R(t) for (2.1) also exists. This is somewhat surprising since S(t) is of course continuously differentiable on (0, ∞); however, it does not seem to be possible to show R ∈ L1loc (⺢+ , B(X)), unless more is assumed on the behaviour of a(z) near zero, or of a(λ) near infinity. For example, a condition like ∞ dr < ∞, for some θ ∈ (π/2, π), ω > ω0 , | a(ω + re±iθ )| r 1 1 is sufficient for integrability of R(t) near zero, − 0 (log t)a(t)dt < ∞ is another one. These are, however, not necessary, as Example 2.3 shows. e) The results in Section 2.6 have been first obtained by Da Prato, Iannelli, and Sinestrari [69] for the case a ∈ BVloc (⺢+ ). There are many other function classes for which maximal regularity results like Theorem 2.4 hold. This subject will be taken up in Sections 7 and 8 again. f) It is instructive to have a look at the fundamental solution φα (t, x) of the problem t u(t, x) = f (t, x) + Γ(α)−1 (t − s)α−1 uxx (s, x)ds, t ≥ 0, x ∈ ⺢, (2.34) 0

66


where α ∈ (0, 2); cp. Example 2.1 in Section 2.3. The fundamental solution represents the resolvent S(t) according to ∞ (S(t)v)(x) = φα (t, x − y)v(y)dy, t ≥ 0, x ∈ ⺢, −∞

and is well-known for α = 1, and for the limiting cases α = 0 and α = 2. φ0 (t, x) = e−|x| /2, √ 2 φ1 (t, x) = e−x /4t / 4πt,

t > 0, x ∈ ⺢;

φ2 (t, x) = (δ0 (t − x) + δ0 (t + x))/2,

t > 0, x ∈ ⺢.

0.5

t > 0, x ∈ ⺢;

α=0

0.4

α=2

α=0.5

0.3

α=1.5 α=1

0.2

0.1

0

-2

-3

-1

0

1

2

3

Figure 2.1: Fundamental solutions φα (1, x) of (2.34)

For general α ∈ (0, 2) it is possible to obtain a representation of φα in terms of the Mittag-Leffler functions M Lβ (z) defined by M Lβ (z) =

∞ k=0

zk , Γ(kβ + 1)

z ∈ ⺓.

This was first observed in Friedman [121] and recently studied in detail by Fujita [124]. The identity α (λ, ξ) = s(λ, ξ 2 ) = (1 + λ−α ξ 2 )−1 /λ φ implies s(t, ξ 2 ) = φα (t, ξ) =

∞ k=0

(−ξ)2k

tkα = M Lα (−ξ 2 tα ), Γ(kα + 1)

t > 0, x ∈ ⺢;


67

thus φα (t, x) can be obtained as the inverse Fourier transform of s(t, ξ 2 ). Note that with ϕα (x) = φα (1, x) we have φα (t, x) = t−α/2 ϕα (xt−α/2 ), and ϕα (x) = (2π)−1

∞ −∞

t > 0, x ∈ ⺢,

eixξ M Lα (−ξ 2 )dξ,

x ∈ ⺢.

Figure 2.1 shows the graphs of φα (1, x) for different values of α. Note that except for α = 2, the support of φα (t, ·) is the whole line; therefore the propagation speed is always infinite, except for α = 2. For α ∈ (0, 1) the maximum stays at x = 0, but as α is increasing beyond 1, this maximum splits into two which are sharpening and moving in different direction to ±1 as α approaches 2. Thus for α ∈ (1, 2) there is a sort of ‘main speed’, this way resembling some properties of the wave equation.

3

Parabolic Equations

As a continuation of Section 2, the concept of parabolicity for Volterra equations of scalar type is introduced and resolvents for such equations are discussed in detail. If the kernel a(t) has some extra regularity property, like convexity, then the resolvent exists, and exhibits the same stability under perturbations as analytic resolvents. Again the maximal regularity property of type C α is valid, even for the perturbed equation. In Section 3.6 we derive a representation formula for the resolvent in case A is the generator of a C0 -semigroup. 3.1 Parabolicity Let X be a complex Banach space, A a closed linear operator in X with dense 1 domain ∞ −εt D(A), and let a ∈ Lloc (⺢+ ) be of subexponential growth, which means e |a(t)|dt < ∞ for all ε > 0. We consider the Volterra equation of scalar 0 type t

u(t) = 0

a(t − τ )Au(τ )dτ + f (t),

t ≥ 0.

(3.1)

If (3.1) admits a bounded resolvent S(t), then by Corollary 1.6 we have a ˆ(λ) ∈ ρ(A) for all λ ∈ ⺓ with Re λ > 0, and H(λ) = (I − a ˆ(λ)A)−1 /λ satisfies |H(λ)| ≤

M , Re λ

Re λ > 0,

(3.2)

for some constant M ≥ 1. If S(t) is even analytic then by Theorem 2.1 we even have M π |H(λ)| ≤ , | arg λ| ≤ + θ, (3.3) |λ| 2 for some θ > 0. The following concept is between these two extremes. Definition 3.1 Equation (3.1) is called parabolic, if the following conditions hold. (P1) a ˆ(λ) = 0, 1/ˆ a(λ) ∈ ρ(A) for all Re λ > 0. (P2) There is a constant M ≥ 1 such that H(λ) = (I − a ˆ(λ)A)−1 /λ satisfies |H(λ)| ≤

M |λ|

for all Re λ > 0.

(3.4)

Thus if (3.1) admits an analytic resolvent S(t) which is bounded on some sector Σ(0, θ) then (3.1) is parabolic; the converse is of course not true. The main difference between (3.3) and parabolicity of (3.1) is that the existence of an analytic continuation of a ˆ(λ) to some sector Σ(0, θ + π/2) is no longer required; recall that (3.3) implies analyticity of a(z) on Σ(0, θ) as we have seen in Corollary 2.3. Definition 3.2 Let a ∈ L1loc (⺢+ ) be of subexponential growth and suppose a ˆ(λ) = 0 for all Re λ > 0. a is called sectorial with angle θ > 0 (or merely θ-sectorial) if | arg a ˆ(λ)| ≤ θ for all Re λ > 0. (3.5) J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_3, © Springer Basel 1993

68

3. Parabolic Equations

69

Here arg a ˆ(λ) is defined as the imaginary part of a fixed branch of log a ˆ(λ), and θ in (3.5) need not be less than π. If a is sectorial, we always choose that branch of log a ˆ(λ) which gives the smallest angle θ; in particular, in case a ˆ(λ) is real for real λ we choose the principal branch. A standard situation leading to parabolic equations is described in Proposition 3.1 Let a ∈ L1loc (⺢+ ) be θ-sectorial for some θ < π, suppose A is closed linear densely defined, such that ρ(A) ⊃ Σ(0, θ), and |(μ − A)−1 | ≤

M |μ|

for all μ ∈ Σ(0, θ).

(3.6)

Then (3.1) is parabolic. Proof: Since 1/ˆ a(λ) ∈ Σ(0, θ) for all Re λ > 0, H(λ) = (I − a ˆ(λ)A)−1 /λ is welldefined, in particular (P1) holds; (3.6) yields (P2). 2

Since a(t) is π2 -sectorial if and only if a(t) is of positive type, we obtain the following nice class of parabolic equations. Corollary 3.1 Let a ∈ L1loc (⺢+ ) be of subexponential growth and of positive type, and let A generate a bounded analytic C0 -semigroup in X. Then (3.1) is parabolic. In the next subsections we show under some additional regularity assumption on the kernel a(t) that parabolic equations admit bounded resolvents which enjoy some further smoothness properties. It will be shown that the maximal regularity result Theorem 2.4 remains true for parabolic equations and that strong perturbation results similar to Theorem 2.3 also hold. For these purposes we first need to discuss the kind of regularity of the kernels which is appropriate for parabolic problems. 3.2 Regular Kernels The notion of regularity needed below is introduced in Definition 3.3 Let a ∈ L1loc (⺢+ ) be of subexponential growth and k ∈ ⺞. a(t) is called k-regular if there is a constant c > 0 such that |λn a ˆ(n) (λ)| ≤ c|ˆ a(λ)|

for all Re λ > 0, 0 ≤ n ≤ k.

(3.7)

Observe that any k-regular kernel a(t), k ≥ 1 has the property that a ˆ(λ) has no zeros in the open right halfplane. In fact, if a ˆ(λ0 ) = 0 then a ˆ(λ) = (λ − λ0 )m ϕ(λ) for some holomorphic function ϕ(λ) with ϕ(λ0 ) = 0; differentiation yields ˆ (λ)−(λ−λ0 )m ϕ (λ) = [(ˆ a (λ)/ˆ a(λ))ϕ(λ)−ϕ (λ)](λ−λ0 )m , m(λ−λ0 )m−1 ϕ(λ) = a hence (3.7) and ϕ(λ0 ) = 0 yield a contradiction.

70


Convolutions of k-regular kernels are again k-regular; this follows easily from the product rule of differentiation. Moreover, integration and differentation are operations which preserve k-regularity as well. On the other hand, sums and differences of k-regular kernels need not be k-regular; consider a(t) = 1 and b(t) = t2 to see this. However, if a(t) and b(t) are k-regular and | arg a ˆ(λ) − arg ˆb(λ)| ≤ θ < π,

Re λ > 0

then a(t) + b(t) is k-regular as well. It is not difficult to verify that (3.7) is equivalent to |(λn a ˆ(λ))(n) | ≤ c |ˆ a(λ)|,

Re λ > 0, 0 ≤ n ≤ k.

(3.8)

and to |λn (log a ˆ(λ))(n) | ≤ c

,

Re λ > 0, 0 ≤ n ≤ k.

(3.9)

where c , c

> 0 are also constants. If a(t) is real-valued and 1-regular then a(t) is sectorial. Indeed,

φ

arg a ˆ(reiφ ) = Im log a ˆ(reiφ ) = Im 0

d log a ˆ(reit )dt = Im dt

hence | arg a ˆ(reiφ )| ≤ cφ ≤

φ 0

a ˆ (reit )ireit dt, a ˆ(reit )

cπ , 2

i.e. a(t) is c π2 -sectorial with c from (3.7). The converse of this is not true as the example a(t) = 1 for t ∈ [0, 1], a(t) = 0 for t > 1 shows. In fact, then a ˆ(λ) = (1 − e−λ )/λ, hence a is π-sectorial but −λˆ a (λ) λ =1− λ , a ˆ(λ) e −1

Re λ > 0,

shows that this function is not bounded in the right halfplane. We shall make use of Lemma 3.1 Suppose g : ⺓+ → ⺓ is holomorphic and satisfies g(λ) = 0, and | arg g(λ)| ≤ θ for Re λ > 0. Then for each n ∈ ⺞ there is a constant cn > 0 depending only on n such that (Re λ)n |g (n) (λ)| ≤ cn θ|g(λ)|,

Re λ > 0.

(3.10)

Proof: Let u(λ) = arg g(λ). The Poisson formula for the halfplane and its analytic completion yield with some constant γ ∈ ⺢ dρ i ∞ 1 − iρλ ]u(iρ) log g(λ) = γ + [ , π −∞ λ − iρ 1 + ρ2


71

hence by differentiation (−1)n (log g(λ))(n) = i

n! π

∞

1 u(iρ)dρ, (λ − iρ)n+1

−∞

Re λ > 0.

This identity implies for λ = σ + iτ σ n |(log g(λ))(n) | ≤ ≤ and from this (3.10) follows.

n! π θ

n! π

∞

σn

−∞ (σ 2 ∞

+ (τ − ρ)2 )

(1 + ρ2 )−

(n+1) 2

n+1 2

|u(iρ)|dρ

dρ,

−∞

2

By means of Lemma 3.1 we can now show that kernels which are sectorial on a sector larger than the right halfplane are k-regular, for every k ∈ ⺞. Proposition 3.2 Suppose a ∈ L1loc (⺢+ ) is such that a ˆ(λ) admits analytic extension to Σ(0, φ), where φ > π2 , and there is θ ∈ (0, ∞) such that | arg a ˆ(λ)| ≤ θ

for all

λ ∈ Σ(0, φ).

(3.11)

Then a(t) is k-regular for every k ∈ ⺞. Proof: Let g : ⺓+ → ⺓ be defined by g(z) = a ˆ(z p ), where p = 2φ π . Then g(z) fulfills the assumptions of Lemma 3.1, hence (3.10) holds. This implies with a ˆ(λ) = g(λα ), α = p1 , λn a ˆ(n) (λ) =

n

bnl g (l) (λα )λαl ,

Re λ > 0, n ∈ ⺞,

l=1

for some constants bnl . Therefore, by Lemma 3.1 we obtain |λn a ˆ(n) (λ)|

≤

n

|bnl ||g (l) (λα )||λα |l

l=1

≤ c|ˆ a(λ)|

n l=1

since with α =

1 p

|bnl |(

|λ|α l ) ≤ c(n)|ˆ a(λ)|, Re λα

Re λ > 0,

0, 0 ≤ n ≤ k − 2, and (−1)k−2 a(k−2) (t) is nonincreasing and convex. Thus by definition a 2-monotone kernel a(t) is nonnegative, nonincreasing and convex, and a(t) is completely monotonic if and only if a(t) is k-monotone for all k ≥ 2. Proposition 3.3 Suppose a ∈ L1loc (⺢+ ) is (k + 1)-monotone, k ≥ 1. Then a(t) is k-regular and of positive type. Proof: Suppose a ∈ L1loc (⺢+ ) is (k + 1)-monotone, for some k ≥ 1. Since 2monotone kernels are of positive type (see below), Lemma 3.1 yields with λ = σ + iρ, σ > 0, ρ ∈ ⺢ σ n |ˆ a(n) (λ)| ≤ cn |ˆ a(λ)|,

Re λ > 0, n ∈ ⺞.

Therefore it is sufficient to prove |ρ|n |ˆ a(n) (λ)| ≤ cn |ˆ a(λ)|,

Re λ > 0, n ≤ k.

Let aσ (t) = a(t)e−σt , t, σ > 0; then aσ (t) is (k + 1)-monotone again, and so by Proposition 3.9 below we have √ |ˆ aσ (iρ)| ≥ ( 2)−3 as well as |ˆ a(n) σ (iρ)| ≤ ck

1 |ρ|

0

1 |ρ|

0

ρ ∈ ⺢, σ > 0,

aσ (t)dt,

tn aσ (t)dt,

ρ ∈ ⺢, σ > 0, n ≤ k,

where ck only depends on k, by Proposition 3.8. These inequalities combined yield (n)

|ρ ||ˆ a n

(λ)| ≤ ck |ρ|

1 |ρ|

n 0

t aσ (t)dt ≤ ck

1 |ρ|

n

0

3

aσ (t)dt ≤ 2 2 ck |ˆ a(λ)|

for all σ = Re λ > 0, ρ ∈ ⺢, and n ≤ k. Therefore a is k-regular. To prove that a(t) is of positive type we use (3.46) for k = 2 and λ = iρ ∞ 1 a ˆ(iρ) = a∞ /iρ + 2 (1 − e−iρt − iρt)da(t), ˙ ρ 0 which gives Re a ˆ(iρ) = ρ−2

0

∞

(1 − cos ρt)da(t), ˙

ρ ∈ ⺢\{0}.

Since a(t) is convex, da(t) ˙ ≥ 0, and so Re a ˆ(iρ) ≥ 0, ρ ∈ ⺢. Re a ˆ(λ) is harmonic in the right halfplane and tends to zero as |λ| → ∞; therefore by the minimum


73

principle we may conclude Re a ˆ(λ) ≥ 0 for all Re λ ≥ 0, i.e. a(t) is of positive type. 2

Another sufficient condition for k-regularity is contained in ∞ k+m,1 Proposition 3.4 Suppose a ∈ Wloc (0, ∞) is such that 0 tl |a(m+l) (t)|dt < ∞ for all 0 ≤ l ≤ k and a(j) (0) = 0 for all 0 ≤ j < m−1; assume there is b ∈ BV (⺢+ ) ˆ with db(λ) = [λm a ˆ(λ)]−1 for all λ > 0. Then a is k-regular. Proof: For Re λ > 0 we have the identity λn

a ˆ(n) (λ) a ˆ(λ)

n+m (n) ˆ = db(λ)λ a ˆ (λ)

ˆ = db(λ) · {[[(−t)n a](n+m) ]∧ (λ) + (tn a(t))(n+m−1) |t=0 } ˆ · = (−1) db(λ) n

n n (n + m)! l=0

l

(m + l)!

{(tl a(m+l) (t))∧ (λ) +

hence λn a ˆ(n) (λ)/ˆ a(λ) is bounded for Re λ > 0, n ≤ k.

m + l l (l+m−1) ta (t)|t=0 }, n+m 2

The cases m = 1 and m = 2 are the most important in applications of Proposition 3.4. 3.3 Resolvents for Parabolic Equations The first main result on resolvents for parabolic equations is the following Theorem 3.1 Let X be a Banach space, A a closed linear operator in X with dense domain D(A), a ∈ L1loc (⺢+ ). Assume (3.1) is parabolic, and a(t) is k-regular, for some k ≥ 1. Then there is a resolvent S ∈ C k−1 ((0, ∞); B(X)) for (3.1), and there is a constant M ≥ 1 such that the estimates |tn S (n) (t)| ≤ M,

for all t ≥ 0, n ≤ k − 1,

|tk S (k−1) (t) − sk S (k−1) (s)| ≤ M |t − s|[1 + log

t ], t−s

0 ≤ s < t < ∞,

are valid. Observe that (3.13) implies 1 s ε |tk S (k−1) (t) − sk S (k−1) (s)| ≤ M (t − s)(1 + ( ) ≤ MTε (t − s)1−ε ε t−s 1−ε for all 0 ≤ s < t < T , i.e. S (k−1) ∈ Cloc ((0, ∞); B(X)) for each ε ∈ (0, 1).

(3.12) (3.13)

74


Proof: The proof of Theorem 3.1 is based on a result on the inversion of the Laplace transform, proved in Section 0.2. Since (3.1) is parabolic by assumption, we have M |H(λ)| ≤ for Re λ > 0. |λ| For H (λ) one obtains H (λ) = −H(λ)/λ + a ˆ (λ)A(I − a ˆ(λ)A)−1 H(λ), and since a(t) is 1-regular this formula gives |H (λ)| ≤

M a ˆ (λ) M (M + 1) M1 + | | ≤ , |λ|2 a ˆ(λ) |λ| |λ|2

Re λ > 0.

Proceeding this way, inductively there follows |H (n) (λ)| ≤

Mn , |λ|n+1

Re λ > 0, n ≤ k,

with some constants Mn . Thus by Proposition 0.1, H(λ) satisfies (H1) and (H2) of Theorem 1.3, and so there is a resolvent S(t) for (3.1). Theorem 0.4 shows moreover that S(t) belongs to C k−1 ((0, ∞); B(X)) and satisfies (3.12) and (3.13). The proof is complete. 2

Combination of Corollary 3.1, Proposition 3.3 and Theorem 3.1 yields Corollary 3.2 Suppose A generates an analytic C0 -semigroup in X, bounded on some sector Σ(0, θ), and let a ∈ L1loc (⺢+ ) be 2-monotone. Then (3.1) admits a α bounded resolvent S(t) in X which moreover belongs to Cloc ((0, ∞); B(X)) for each α < 1. Proof: By Proposition 3.3, a(t) is 1-regular and of positive type, and therefore (3.1) is parabolic by Corollary 3.1. Theorem 3.1 with k = 1 yields the assertion. 2

In Section 1.3, Corollary 1.2, we have seen that (3.1) is well-posed if A is negative semidefinite in the Hilbert space X, and a ∈ BVloc (⺢+ ) is such that the measure da is of positive type. This is in general a hyperbolic situation; however, we can prove now the following parabolic extension of this result. Corollary 3.3 Suppose A generates a C0 -semigroup T (z) which is analytic in Σ(0, π2 ), and uniformly bounded in each subsector Σ(0, θ). Let a ∈ BVloc (⺢+ ) be of the form t

a(t) = a0 +

0

a1 (s)ds,

t > 0,

(3.14)


75

where a0 ≥ 0, a1 ∈ L1 (⺢+ ) is 3-monotone, and t t−1 0 τ a1 (τ )dτ limt→0 < ∞. t a0 + 0 −τ a˙ 1 (τ )dτ

(3.15)

Then there exists a bounded resolvent S(t) for (3.1) in X, which moreover belongs 1+α to Cloc ((0, ∞); B(X)) for each α < 1, and tS (t) is bounded on (0, ∞). Observe that (3.15) holds if a0 > 0 or if limt→0 − ta˙ 1 (t)/a1 (t) > 0; on the other hand (3.15) implies a0 > 0 or a1 (0+) = ∞, i.e. da(t) must have a singularity at t = 0. Proof: The assumption on A implies σ(A) ⊂ (−∞, 0] and (3.6) for every θ < π. For parabolicity of (3.1) it is therefore sufficient to show that a(t) is θ-sectorial for some θ < π. By Proposition 3.1, a ˆ(λ) is of the form a ˆ(λ) = λ−1 (a0 + a ˆ1 (λ)),

Re λ > 0.

and so we only need to show | arg(a0 + a ˆ1 (λ))| ≤ θ0
0.

(3.16)

Since a1 (t) is 3-monotone, it is 2-regular and of positive type, by Proposition 3.3; the remarks following Definition 3.3 then show that a(t) is 2-regular again. Since a1 (t) is 3-monotone, Proposition 3.10 yields 3 |ρ| 5

1 |ρ|

ta1 (t)dt ≤ |Im a ˆ1 (iρ)| ≤ 12|ρ|

0

1 |ρ|

0

ta1 (t)dt,

ρ ∈ ⺢,

as well as 3 5

1 |ρ|

0

−ta˙ 1 (t)dt ≤ Re a ˆ1 (iρ) ≤ 4

0

1 |ρ|

−ta˙ 1 (t)dt,

ρ ∈ ⺢,

and therefore (3.16) is equivalent to φ(ρ) = |ρ|

1 |ρ|

0

ta1 (t)dt/(a0 +

1 |ρ|

0

−ta˙ 1 (t)dt) ≤ C,

ρ ∈ ⺢;

(3.17)

apply the maximum principle to the harmonic function arg(a0 +ˆ a1 (λ)) in the right halfplane ⺓+ to see this. φ(ρ) is a continuous function on ⺢\{0}, therefore (3.17) needs only to be checked at ρ = 0 and ρ = ∞. Condition (3.15) shows that (3.17) holds at ρ = ∞; the estimate ∞ 1 φ(ρ) ≤ a1 (t)dt/(a0 + −τ a˙ 1 (τ )dt) < ∞ for |ρ| ≤ 1 0

yields (3.17) near ρ = 0.

0

76


Thus Proposition 3.1 shows that (3.1) is parabolic. As observed already, a(t) is 2-regular, and so Theorem 3.1 yields the assertion. 2

It should be observed that in Corollary 3.3 the condition “a1 ∈ L1 (⺢+ )” can be −1 replaced by “0 ∈ ρ(A)” since a ˆ(λ) → 0 as λ → 0, and so (3.17) then is not needed at ρ = 0. 3.4 Perturbations Consider as in Section 2.5 the perturbed equation u(t) = f (t) + (a + a ∗ dk) ∗ Au(t) + b ∗ u(t),

t ≥ 0,

(3.18)

where a(t) is such that (3.1) is parabolic, b ∈ L1loc (⺢+ ), and k ∈ BVloc (⺢+ ) is continuous at zero. If a(t) is in addition 1-regular then by Theorem 3.1, Equation (3.1) admits a resolvent. It turns out that in this situation (3.18) is well-posed as well. This shows that for parabolic equations a perturbation theorem like in the case of analytic resolvents is valid, provided a(t) is in addition 1-regular. Theorem 3.2 Suppose (3.1) is parabolic, a ∈ L1loc (⺢+ ) is 1-regular, and let b ∈ L1loc (⺢+ ), k ∈ BVloc (⺢+ ) such that k(0) = k(0+). Then (3.18) admits a resolvent S(t). Proof: Without loss of generality we may assume that b(t) and dk(t) are Laplace transformable, and since k(t) is continuous at zero, we may even assume ˆ |ˆb(λ)| + |dk(λ)| ≤η

for Re λ ≥ 0,

(3.19)

where η > 0 is any prescribed number; multiply (3.18) with e−ωt with some sufficiently large ω, to see this. As in the proof of Theorem 2.4 we let r(t) and l(t) be defined by r = b + b ∗ r, l = k + 1 ∗ r + k ∗ r. (3.20) The resolvent of (3.18) then must satisfy the equation S = I + Aa ∗ S + dk ∗ Aa ∗ S + b ∗ S,

(3.21)

hence as in the proof of Theorem 2.4 ∞ 1 n ˆ ˆ ˆ S(λ) = (1 − ˆb(λ) − (1 + dk(λ))ˆ a(λ)A)−1 = (1 + rˆ(λ)) [dl(λ)] Hn (λ), λ n=0

for Re λ > 0, where Hn (λ) =

1 n a ˆ(λ) An (1 − a ˆ(λ)A)−(n+1) , λ

Re λ > 0.

(3.22)


77

Since (3.1) is parabolic we obtain |Hn (λ)| ≤ |H0 (λ)||ˆ a(λ)A(I − a ˆ(λ)A)−1 |n ≤

M0 · M n, |λ|

Re λ > 0,

for some constants M0 , M ≥ 1. 1-regularity of a(t) and 1 a ˆ (λ) a ˆ (λ) Hn (λ) = − Hn (λ) + n Hn (λ) + (n + 1) Hn+1 (λ), λ a ˆ(λ) a ˆ(λ) yields the estimate |Hn (λ)| ≤

M0 M n M0 M n M0 M n+1 M1 M n (n + 1) + nC + (n + 1)C ≤ . |λ|2 |λ|2 |λ|2 |λ|2

α Therefore, there are functions Sn ∈ L∞ (⺢+ ; B(X))∩Cloc ((0, ∞); B(X)), α ∈ (0, 1), such that Sˆn (λ) = Hn (λ) for Re λ > 0,

and |Sn (t)| ≤ M2 · M n (n + 1),

t > 0,

for some constant M2 which is independent of n, by Theorem 0.4. Define S(t) = (δ + r) ∗

∞

[dl]∗n ∗ Sn (t),

t > 0;

(3.23)

n=0

this series converges absolutely if η > 0 is sufficiently small and it is clear from the construction that (3.21) holds. Each function Sn (t) commutes with A, and so (3.21) evaluated at x ∈ D(A) shows that S(t)x is continuous on ⺢+ , hence S(t) is strongly continuous on ⺢+ , by the Banach-Steinhaus Theorem. Thus S(t) is the resolvent for (3.18) and the proof is complete. 2

It should be noted that in case a(t) is k-regular, k > 1, then Sn (t) defined in the proof of Theorem 3.2 belongs to C k−1 ((0, ∞); B(X)); however, S(t) in general will only be continuous, since the discontinuities in l(t) will be reproduced in S(t). No matter how smooth a(t) is, the resolvent S(t) can be only as smooth as l(t). 3.5 Maximal Regularity Consider again the perturbed equation (3.18) where a(t) is such that (3.1) is parabolic, b ∈ L1loc (⺢+ ), and k ∈ BVloc (⺢+ ) is continuous at zero. It is a surprising fact that (3.18) has the maximal regularity property of type C α in case a(t) is 2regular, although the resolvent S(t) of (3.18) is merely continuous. Theorem 3.3 Suppose a ∈ L1loc (⺢+ ) is 2-regular and such that (3.1) is parabolic, let b ∈ BVloc (⺢+ ), and k ∈ BVloc (⺢+ ) with k(0) = k(0+), let J = [0, T ] and α ∈ (0, 1). Then (i) for each f ∈ C0α (J; X) there is a mild solution u ∈ C0α (J; X) of (3.18);

78


(ii) for each f ∈ C0α (J; XA ) there is a strong solution u ∈ C0α (J; XA ) of (3.18); (iii) for each f = a ∗ g, g ∈ C0α (J; X) there is a strong solution u ∈ C0α (J; XA ) of (3.18). Proof: It is sufficient to prove (i); (ii) and (iii) follow from (i) as in the proof of Theorem 2.4. According to the proof of Theorem 3.2, the resolvent S(t) for (3.18) is of the form ∞ S(t) = (δ + r) ∗ [dl]∗n ∗ Sn (t), t > 0, (3.24) n=0

where r(t) and l(t) are defined as in (3.20), and w.l.o.g. ∞ ∞ |r(t)| + |dl(t)| ≤ η, 0

(3.25)

0

η > 0 being any prescribed number. Sn (t) satisfies 1 n Sˆn (λ) = Hn (λ) = a ˆ(λ) An (1 − a ˆ(λ)A)−(n+1) , λ

Re λ > 0.

Since a(t) is by assumption 2-regular, as in the proof of Theorem 3.2 there are constants M0 and M such that |Hn (λ)| + |λHn (λ)| + |λ2 Hn

(λ)| ≤

M0 n M , |λ|

Re λ > 0, n ∈ ⺞.

(3.26)

Theorem 0.4 then shows Sn ∈ C 1 ((0, ∞); B(X)) with |Sn (t)| + |tSn (t)| ≤ M1 M n ,

t>0

(3.27)

and |t2 Sn (t) − s2 Sn (s)| ≤ M1 M n (t − s)(1 + log

t ), t−s

0 ≤ s < t < ∞.

(3.28)

The latter yields the estimate |Sn (t + h) − Sn (t)| ≤ M2 M n

h t (1 + log(1 + )) t(t + h) h

for all t, h > 0. (3.29)

Given f ∈ C0α (J; X) we set un (t) =

d (Sn ∗ f )(t) = Sn (t)f (t) + dt

0

t

Sn (t − τ )(f (τ ) − f (t))dτ,

t ∈ J, (3.30)

and follow the proof of Theorem 2.4 to obtain un ∈ C0α (J; X) as well as the estimate |un |α ≤ CM n |f |α , n ∈ ⺞0 , (3.31)


79

where C is a constant independent of n; observe that the estimate of I4 there has to be modified slightly using (3.29). Then set u(t) = (δ + r) ∗

∞

[dl]∗n ∗ un (t),

t ∈ J;

n=0

the series converges absolutely in C0α (J; X) provided ηM < 1, and we obtain ∞

|u|α ≤ (1 + η)C

(ηM )n |f |α = C1 |f |α .

n=0

If f ∈ C 1 (J; X), f (0) = 0 then un = Sn ∗ f implies that u = S ∗ f is the unique mild solution of (3.18). The result now follows from an approximation argument like in proof of Theorem 2.4. 2

Corollaries analogous to Corollaries 2.5 and 2.6 can be obtained as well; we leave this to the reader. 3.6 A Representation Formula Let a ∈ L1loc (⺢+ ) be 1-regular and sectorial with angle θ < π2 and suppose A generates a bounded C0 -semigroup T (τ ) in the Banach space X. Then ρ(A) ⊃ ⺓+ and there is a constant M1 ≥ 1 such that |(μ − A)−1 | ≤

M1 Re μ

for Re μ > 0,

in particular, (3.6) holds on each sector Σ(0, θ) with θ < π2 . By Proposition 3.1 this implies that (3.1) is parabolic, and Theorem 3.1 shows the existence of a bounded resolvent S(t) for (3.1). In this section we want to derive a representation formula for S(t) in terms of the semigroup T (τ ) and certain functions wt ∈ L1 (⺢+ ). To see how this becomes possible recall that ∞ (I − μA)−1 = μ−1 e−τ /μ T (τ )dτ, Re μ > 0; (3.32) 0

thus we obtain ˆ S(λ) = H(λ) = λ−1 (I − a ˆ(λ)A)−1 =

i.e.

1 λˆ a(λ)

∞

e−τ /â(λ) T (τ )dτ,

0

∞

ˆ S(λ) =

h(λ, τ )T (τ )dτ,

Re λ > 0,

(3.33)

Re λ > 0, τ ≥ 0.

(3.34)

0

where h(λ; τ ) =

e−τ /â(λ) , λˆ a(λ)

80


The Laplace transform of h(λ; τ ) w.r.t. τ is then given by σ(λ; μ) =

1 , λ(1 + μˆ a(λ))

Re λ > 0, Re μ > 0,

(3.35)

which on the other hand is the Laplace transform of the function s(t; μ) introduced in Section 1.3 w.r.t. t. Next we show that s(t; μ) is the Laplace transform w.r.t. τ of an L1 -function wt (τ ). Proposition 3.5 Suppose the kernel a ∈ L1loc (⺢+ ) is 1-regular and sectorial with angle θ < π2 , and let s(t; μ) denote the solution of s(t) + μ(a ∗ s)(t) = 1,

t ≥ 0, μ ∈ ⺓.

1,1 Then there is a uniformly bounded family {wt }t>0 ⊂ BV (⺢+ ) ∩ Wloc (⺢+ ) such that ∞ s(t; μ) = − e−μτ w˙ t (τ )dτ, t > 0, Re μ > 0. (3.36) 0

∗

The function w˙ : (0, ∞) → L1 (⺢+ ) is continuous and wt −e0 in BV (⺢+ ) as t → 0, where e0 (t) denotes the Heaviside function. Proof: Fix t > 0 and ε > 0; then s(t; μ) is represented as eεt ∞ iρt s(t; μ) = e σ(ε + iρ, μ)dρ, t > 0, Re μ ≥ 0, 2π −∞

(3.37)

by inversion of the Laplace transform. Integration by parts and a differentation w.r.t. μ yields ∂ eεt ∞ iρt ∂ 2 − s(t; μ) = e σ(ε + iρ; μ)dρ. (3.38) ∂μ 2πt −∞ ∂λ∂μ By a simple computation we obtain 2

∂2 1 a ˆ(λ) λˆ a (λ) λˆ a (λ) μˆ a(λ) σ(λ; μ) = 2 [ (1 − ) + 2 ], 2 ∂λ∂μ a(λ)) a ˆ(λ) a ˆ(λ) (1 + μˆ a(λ))3 λ (1 + μˆ and a direct estimate which employs 1-regularity and θ < |

C ∂2 |ˆ a(λ)| σ(λ; μ)| ≤ · 2 ∂λ∂μ |λ| (1 + |μ||ˆ a(λ)|)2

π 2

gives

for Re λ > 0, Re μ ≥ 0,

(3.39)

where C > 0 denotes a constant independent of μ and λ. Combining (3.38) and (3.39) yields ∞ ∂ dρ eεt ∞ ∞ |ˆ a(ε + iρ)| | s(t; iτ )|dτ ≤ 2C · 2 dτ 2 2πt 0 a(ε + iρ)|) ε + ρ2 −∞ ∂μ −∞ (1 + τ |ˆ ∞ e eεt ∞ dr dρ C eεt =C = 2C ( ) 2 = 2 2 2πt −∞ 0 (1 + r) ε + ρ π εt π


81

with the choice ε = 1t . Thus for fixed t > 0, s(t; ·) belongs to the Hardy space ∂ H ∞ (⺓+ ) and ∂μ s(t; ·) ∈ H 1 (⺓+ ); by Hardy’s inequality (see Duren [106]), there are functions v t ∈ L1 (⺢+ ) such that vt (μ) = s(t; μ) for Re μ > 0, and |v t |1 ≤ Ce 2π , for all t > 0. Define ∞ wt (τ ) = v t (s)ds, t > 0, τ ≥ 0; τ

then w˙ t (τ ) = −v t (τ ), hence (3.36) is valid, and the family {wt }t>0 ⊂ BV (⺢+ ) is uniformly bounded. A similar argument involving Hardy’s theorem shows that the function w˙ : α (0, ∞) → L1 (⺢+ ) is continuous (even Cloc ((0; ∞); L1 (⺢+ )) for any α ∈ (0, 1)) and t ∗ w −e0 follows from s(t; μ) → 1 as t → 0 for every μ ∈ ⺓, since span{eμτ : μ < 0} is dense in C0 (⺢+ ). 2

Since w˙ : (0, ∞) → L1 (⺢+ ) is continuous and bounded, the vector valued Laplace ˙ transform w(λ) exists for Re λ > 0. The ordinary Laplace transform L : L1 (⺢+ ) → H ∞ (⺓+ ) is linear and bounded, hence ˙ Lw(λ) = L w(λ) ˙ = −ˆ s(λ; μ) = −σ(λ; μ) holds for Re λ > 0 and Re μ ≥ 0. Since on the other hand σ(λ; ·) = Lh(λ, ·)

for Re λ > 0,

we obtain from uniqueness of the ordinary Laplace transform ˙ w(λ) = −h(λ; ·),

Re λ > 0.

(3.40)

Via (3.33) this yields ˆ S(λ) = 0

=

∞

−(

h(λ; τ )T (τ )dτ = − ∞

∞

˙ w(λ; τ )T (τ )dτ

0

w˙ t (τ )T (τ )dτ )∧ (λ),

0

and so by uniqueness of the vector-valued Laplace transform we arrive at the representation formula ∞ S(t) = − w˙ t (τ )T (τ )dτ, t > 0, (3.41) 0

we have been looking for. 3.7 Comments a) The concept of parabolicity employed in this section has been introduced in Pr¨ uss [278] as well as the notion of k-regular kernels, and the results of Sections 3.2

82


and 3.3 appear also there. The perturbation theorem, Theorem 3.2, and Theorem 3.3 on maximal regularity are new, as is the representation formula in Section 3.6 for the case considered here. b) In the situation of Proposition 3.1 one can show without further assumptions that there is an Lploc -resolvent S(t) for (3.1), i.e. a function S ∈ Lploc (⺢+ ; B(X)), ˆ for all 1 ≤ p < ∞, such that S(λ) = H(λ), Re λ > 0, and the resolvent equation holds a.e. The proof of this result relies on the Dunford integral representation −1 (μ + A)−1 s(t; μ)dμ, t > 0, S(t) = 2πi Γ where Γ is an appropriate contour in the complex plane, and on Lp -estimates for the scalar function s(·; μ). The latter in turn can be obtained from standard estimates in the theory of H p -spaces, and from lower bounds for the Laplace transform of sectorial kernels. This is similar to the method used t by Friedman and Shinbrot [123] and Friedman [121] for the case a(t) = a0 + 0 a1 (s)ds with 1,1 a1 ∈ Wloc (⺢+ ). However, the latter result is a special case of Theorem 3.2, even 1 a1 ∈ Lloc (⺢+ ) is sufficent. c) 2- and 3-monotone kernels have been used extensively in the theory of scalar and vector-valued Volterra equations; see e.g. Levin [213], Shea and Wainger [302], Carr and Hannsgen [38], [39], Hannsgen [163], [162], Kiffe and Stecher [196], and the recent monograph by Gripenberg, Londen, and Staffans [156], as well as the references given there. d) The estimates on a ˆ(n) in the appendix are generalizations of the results of Shea and Wainger [302], Hannsgen [161], Carr and Hannsgen [38] for 2-monotone kernels; cp. also Gripenberg, Londen, and Staffans [156]. e) For parabolic equations perturbations A + B, where D(B) ⊃ D(A), and for each ε > 0 there is C(ε) > 0 such that |Bx| ≤ ε|Ax| + C(ε)|x|,

x ∈ D(A),

are again possible, if one applies an exponential shift. Parabolicity is in this sense invariant under such perturbations. f) It is also possible to obtain spatial regularity of the resolvent S(t) for equations of parabolic type, provided a(t) is k-regular, for some large enough k. In fact, after an exponential shift we obtain by the arguments in the proof of Theorem 3.1 and by Lemma 8.1 the estimate |AH (k) (λ)| ≤ M/(|λ|2 + 1),

Re λ ≥ 0,

for some sufficiently large k. This yields |tk AS(t)| ≤ Ceωt ,

t ≥ 0,

i.e. R(S(t)) ⊂ D(A) for all t > 0; cp. Section 2.4.


83

g) Observe that Proposition 3.5 yields an estimate of the form |s(t; μ)| ≤ M

for all t > 0, Re μ > 0,

provided a(t) is 1-regular and θ-sectorial with θ < π/2. The family w(t; τ ) = wt (τ ) constructed there satisfies Var w(t; ·)|∞ 0 ≤ M,

for all t > 0.

It seems to be unknown whether in this situation also Var w(·; τ )|∞ 0 ≤ M, holds; the latter would imply ∞ M

|r(t; μ)|dt ≤ |μ| 0

for all τ > 0

for all t > 0, Re μ > 0.

However, this in turn can be proved at least if a(t) is in addition 2-monotone. It will be seen that the family w(t; τ ) is of central importance for the theory presented in Section 4. Appendix: k-monotone Kernels A kernel a ∈ L1loc (⺢+ ) is called 1-monotone if a(t) is nonnegative and nonincreasing; a is called k-monotone (k ≥ 2) if a ∈ C k−2 (0, ∞) satisfies (−1)n a(n) (t) ≥ 0 for all t > 0, n ≤ k − 2, and (−1)k−2 a(k−2) (t) is nonincreasing and convex. To keep the presentation self-contained we now prove some estimates for k-monotone kernels which have been used above. The behaviour of such kernels at t = 0 is described in Proposition 3.6 Let a ∈ L1loc (⺢+ ) be k-monotone for some k ≥ 1. Then (i) a(n) (t)tn+1 → 0 as t → 0, n ≤ k − 1; 1 (ii) a(n) (t)tn ∈ L1 (0, 1) for all n ≤ k − 1, and 0 (−t)k da(k−1) (t) < ∞; (iii) For t > 0, l ∈ ⺞0 we have the identity 0

t

τl tj+l+1 a(τ )dτ = (−1)j a(j) (t) + l! (j + l + 1)! k−1 j=0

t 0

τ k+l (−1)k (k−1) da (τ ). (k + l)!

(3.42)

Proof: Since a(t) is nonincreasing and nonnegative and belongs to L1 (0, 1), we have t ta(t) ≤ a(s)ds → 0 as t → 0, 0

as well as

t

0≤ ε

t

−τ a(τ ˙ )dτ = −ta(t) + εa(ε) + ε

a(τ )dτ → −ta(t) +

t

a(τ )dτ 0

84


as ε → 0; this proves (i) and (ii) for n = 0, n = 1 respectively. Suppose (ii) holds for some n ≤ k − 1; then t n t n tn+1 τ τ n (n) n (n) 0≤ (−1) a (t) = ( dτ )(−1) a (t) ≤ (−1)n a(n) (τ )dτ → 0 (n + 1)! n! 0 0 n! as t → 0, since (−1)n a(n) (t) is nonincreasing and tn a(n) (t) ∈ L1 (0, 1). Thus (i) holds for n as well and the relation t τ n+1 (n+1) 0 ≤ (−1)n+1 a (τ )dτ (n + 1)! ε t tn+1 (n) εn+1 (n) τn = (−1)n+1 a (t) + (−1)n a (ε) + (−1)n a(n) (τ ) (n + 1)! (n + 1)! n! ε shows that (ii) also holds for n + 1. Thus (i) and (ii) follow by induction. (iii) is obtained by integration by parts. 2

The next proposition contains the behaviour of a(t) at t = ∞. Proposition 3.7 Let a ∈ L1loc (⺢+ ) be k-monotone for some k ≥ 1, and let a∞ = limt→∞ a(t). Then (i) a(n) (t)tn → 0 as t → ∞, 1 ≤ n ≤ k − 1; ∞ (ii) a(n) (t)tn−1 ∈ L1 (1, ∞), 1 ≤ n ≤ k − 1, and 1 tk−1 (−1)k da(k−1) (t) < ∞; (iii) For t > 0, l ≤ k − 1 we have the identity ∞ l+1 l tl−j+1 kτ (k−1) (−1) da (τ ) = (−1)k−j a(k−j) (t) − δl,k−1 a∞ . (3.43) l! (l − j + 1)! t j=1 Proof: Since a(t) is nonincreasing and nonnegative, a∞ = limt→∞ a(t) exists and is nonnegative; from R 0≤ −a(τ ˙ )dτ = a(t) − a(R) → a(t) − a∞ , as R → ∞ t

there follows a˙ ∈ L1 (1, ∞). Suppose a(n) (t)tn−1 ∈ L1 (1, ∞) for some 1 ≤ n ≤ k−1; then t tn τ n−1 0 ≤ (−1)n a(n) (t) (1 − 2−n ) = (−1)n a(n) (t) dτ t (n − 1)! n! 2 t τ n−1 ≤ (−1)n a(n) (τ )dτ → 0 as t → ∞, t (n − 1)! 2 since (−1)n a(n) (t) is nonincreasing, hence a(n) (t)tn → 0 as t → ∞. Furthermore, the identity R τn 0 ≤ (−1)n+1 a(n+1) (τ )dτ n! t R n R (n) tn τ n−1 (n) = (−1)n+1 a (R) + (−1)n a(n) (t) + (−1)n a (τ )dτ n! n! (n − 1)! t


85

shows tn a(n+1) (t) ∈ L1 (1, ∞). Thus (i) and (ii) follow by induction. (iii) is obtained by integration by parts. 2

Next we consider the Laplace-transform of a k-monotone kernel a(t). An integration by parts shows ∞ λˆ a(λ) = a∞ + −a(t)(1 ˙ − e−λt )dt, Re λ > 0; (3.44) 0

thus by Propositions 3.6 and 3.7 we conclude that λˆ a(λ) admits continuous extension to ⺓+ if a(t) is merely 1-monotone, and (3.44) holds for all Re λ ≥ 0. Furthermore, ε ∞ 1 − e−λt |λˆ a(λ)| ≤ a∞ + −a(t)t| ˙ |dt + −a(t)|1 ˙ − e−λt |dt t 0 ε ∞ ε ≤ a∞ + |λ| −a(t)tdt ˙ +2 −a(t)dt ˙ 0 ε ε = a∞ + |λ|(−εa(ε) + a(t)dt) + 2(a(ε) − a∞ ), 0

−z

since | 1−ez

| ≤ 1 for Re z ≥ 0. Hence with ε = 1/|λ|,

|λˆ a(λ)| ≤ |λ|

1 |λ|

a(t)dt + a(1/|λ|) − a∞ ≤ 2|λ|

0

1 |λ|

a(t)dt,

0

i.e. we have the upper estimate |ˆ a(λ)| ≤ 2

1 |λ|

0

a(t)dt,

Re λ ≥ 0.

(3.45)

Integrating (k − 1)-times by parts in (3.44), when a(t) is k-monotone we obtain by means of Propositions 3.6 and 3.7 k

(−λt)j − e−λt )da(k−1) (t), j!

(3.46)

∞ k−1 (k − 1)! (−z)j (k − 1)!j! (−z)j −z k−1 ( − e ) = (−1) · , zk j! (j + k)! j! j=0 j=0

(3.47)

λ a ˆ(λ) = λ

a∞ −

∞ k−1

Re λ ≥ 0.

k−1

(

0

j=0

With fk (z) =

(3.46) can be rewritten as a ˆ(λ) =

a∞ − λ

∞ 0

fk (λt)

tk da(k−1) (t), (k − 1)!

Re λ ≥ 0,

(3.48)

86


and this gives the following representation of aˆ(n) (λ). a ˆ(n) (λ) =

(−1)n n!a∞ − λn+1

∞

(n)

fk (λt)tn+k

0

da(k−1) (t) , (k − 1)!

Re λ ≥ 0.

(3.49)

(3.49) can be used to obtain estimates for a ˆ(n) (λ), n ≤ k − 1; for this purpose we have to study the functions fk (z) in detail. Lemma 3.2 Let k ≥ 1 and fk (z) be defined by (3.47). Then 1 (i) fk (z) = 0 (t − 1)k−1 e−tz dt, for all k ≥ 1, z ∈ ⺓; 1 (n) (ii) |fk (z)| ≤ 0 (1 − t)k−1 tn dt = n!(k−1)! (n+k)! , k ≥ 1, n ≥ 0, Re z ≥ 0; (n)

(iii) |fk (z)| ≤

n! 2n+1 , 0 |z|n+1

1

Proof: Let gk (z) = have

0

≤ n ≤ k − 1, Re z ≥ 0.

(t − 1)k−1 e−tz dt; we want to show fk = gk . For k = 1 we

g1 (z) =

1

e−tz dt =

0

1 (1 − e−z ) = f1 (z). z

Next we observe that 1 −tz k 1 k −tz ke 1 gk+1 (z) = (t − 1) e dt = (t − 1) | + (t − 1)k−1 e−tz dt −z 0 z 0 0 =

(−1)k k + gk (z) z z

for all k ≥ 1, z ∈ ⺓.

On the other hand, we also have fk+1 (z) =

k (−1)k fk (z) + z z

for all k ≥ 1, z ∈ ⺓,

hence gk and fk satisfy the same recursion, and so fk = gk , by f1 = g1 ; this proves (i). (n) To prove (ii) we simply estimate fk (z), which is given by

(n)

fk (z) = to the result (n) |fk (z)|

≤

1 0

(t − 1)k−1 (−t)n e−tz dt,

1 0

(1 − t)k−1 tn dt =

(3.50)

n!(k − 1)! . (n + k)!

To prove (iii), integrate (n + 1)-times by parts in (3.50); this yields (n)

fk (z) =

(−1)n z (n+1)

1 0

{(t − 1)k−1 tn }n+1 e−tz dt −

(−1)n {(t − 1)k−1 tn }(n) e−tz |10 , z n+1


87

since the integrand vanishes at t = 0, 1 to the order n. We have (

n+1 n + 1 tl−1 (t − 1)k−1−l d n+1 ) {(t − 1)k−1 tn } = · n!(k − 1)!, dt l (l − 1)! (k − 1 − l)! l=1

and the term with l = n + 1 is absent in case n = k − 1. This yields (n) |fk (z)|

n+1 n + 1 1 tl−1 (1 − t)k−1−l 1 ≤ n+1 {n! + n!(k − 1)! dt} |z| l 0 (l − 1)! (k − 1 − l)! l=1

=

n+1 n + 1 n! 2n+1 n! {1 + }= n+1 |z| l |z|n+1

for n < k − 1,

l=1

and similarly also for n = k − 1. 2

Lemma 3.2 now yields the desired estimates for a ˆ(n) (λ). From (3.49) we obtain n!a∞ + |λ|n+1

|ˆ a(n) (λ)| ≤ ≤

n!a∞ +n! |λ|n+1

1 |λ|

+

∞ 0

(n)

|fk (λt)|

tn+k (−1)k da(k−1) (t) (k − 1)!

tn+k n!2n+1 (−1)k da(k−1) (t)+ n+1 (n + k)! |λ|

0

n!a∞ ≤ + |λ|n+1

1 |λ|

0

tn a(t)dt − n!

k−1 j=0

n+1 k−1

n!2 |λ|n+1

j=0

1 |λ|

(−1)j (j) n!2n+1 a (1/|λ|)|λ|−j − a∞ j! |λ|n+1

tn a(t)dt + n!(2n+1

0

= 2n+1

(k + n)! (k − 1)!

1 |λ|

tk−1 (−1)k da(k−1) (t) (k − 1)!

(−1)j a(j) (1/|λ|)(|λ|)−(j+n+1) (j + n + 1)!

by Lemma 3.2 and Propositions 3.6 and 3.7. Since application of Proposition 3.6 finally yields |ˆ a(n) (λ)| ≤

∞

1 |λ|

tn a(t)dt,

0

(j+n+1)! j!

(k + n)! − 1) (k − 1)!

0

1 |λ|

≤

(k+n)! (k−1)!

another

tn a(t)dt n!

Re λ ≥ 0, n ≤ k − 1.

Summarizing we have Proposition 3.8 Let a ∈ L1loc (⺢+ ) be k-monotone for some k ≥ 1. Then |ˆ a(n) (λ)| ≤ 2n+1

(2n + 1)! n!

1 |λ|

0

tn a(t)dt,

Re λ ≥ 0, 0 ≤ n ≤ k − 1.

(3.51)

88


To derive lower bounds for a ˆ(λ) we write λ = σ + iρ; then √ 2|ˆ a(λ)| ≥ Re a ˆ(λ) − Im a ˆ(λ) =

π 2|ρ|

= 0

aσ (t) cos ρtdt +

≥ (cos 1)

0

1 |ρ|

∞

aσ (t) cos ρtdt +

0

∞ 0

(aσ (t) − aσ (t +

σ 1 aσ (t)dt ≥ e− |ρ| 2

1 |ρ|

∞

0

aσ (t) sin ρtdt

π )) sin ρtdt ρ

a(t)dt.

0

if a(t) is nonnegative and convex; observe that a(t) is convex if and only if a(t) − a(t + h) is nonincreasing in t, for every h > 0. This yields Proposition 3.9 Let a ∈ L1loc (⺢+ ) be 2-monotone. Then 2 0

1 |ρ|

√ a(t)dt ≥ |ˆ a(iρ)| ≥ (2 2)−1

1 |ρ|

0

a(t)dt,

for all ρ = 0.

(3.52)

Observe that the constants appearing in Propositions 3.8 and 3.9 do not depend on the particular kernel in question, they only depend on the property of k-monotonicity. Next we derive estimates for Re a ˆ(iρ) and −Im a ˆ(iρ) from above and below. Proposition 3.10 Let a ∈ L1loc (⺢+ ) be 2-monotone. Then 1 1 (i) 35 ρ 0ρ ta(t)dt ≤ −Im a ˆ(iρ) ≤ 12ρ 0ρ ta(t)dt, ρ > 0; 1 (ii) Re a ˆ(iρ) ≤ 4 0ρ −ta(t)dt, ˙ ρ > 0. 1 (iii)If a is also 3-monotone then Re a ˆ(iρ) ≥ 35 0ρ −ta(t)dt, ˙ ρ > 0. Proof: By (3.46) with k = 2 we have a∞ 1 a ˆ(iρ) = − iρ (iρ)2

∞ 0

(1 − iρt − e−iρt )da(t), ˙

hence with 0 ≤ t − sin t ≤ min(1 + t, t3 /3!) ∞ a∞ 1 −Im a ˆ(iρ) = + 2 (ρt − sin ρt)da(t) ˙ ρ ρ 0 ∞ ρ1 a∞ 1 1 (ρt)3 ≤ + 2 (ρt + 1)da(t) ˙ + 2 da(t) ˙ (3.53) ρ ρ ρ1 ρ 0 3! ρ1 ρ1 3 a∞ 2 1 1 1 t = − 2 a( da(t) ˙ ≤ 12ρ ˙ 2 ) + (a( ) − a∞ ) + ρ ta(t)dt ρ ρ ρ ρ ρ 0 3! 0


89 3

2

3

t t by Proposition 3.6. Similarly, with t − sin t ≥ t3! (1 − 20 ) ≥ 10 for t ∈ [0, 1], we obtain 1 ∞ a∞ 6ρ ρ t3 1 −Im a ˆ(iρ) ≥ + da(t) ˙ + 2 (ρt − sin ρt)da(t) ˙ ρ 10 0 3! ρ ρ1

ρ1 7a∞ 6ρ 1 1 1 1 3 1 7 1 + [ 3 a( ˙ ) − 2 a( ) + ta(t)dt] + a( ) − (1 − sin 1)a( ˙ ) 2 10ρ 10 6ρ ρ 2ρ ρ 10ρ ρ 10ρ ρ 0 ρ1 ρ1 7a∞ 6 3 ≥ + ρ ta(t)dt ≥ ρ ta(t)dt, 10ρ 10 0 5 0 since 7(1 − sin 1) ≥ 1; this proves (i). On the other hand, with 1 − cos t ≤ min(t2 /2, 2) ∞ ∞ ρ1 2 1 2 t Re a ˆ(iρ) = 2 (1 − cos ρt)da(t) ˙ ≤ 2 da(t) ˙ + da(t) ˙ ρ 0 ρ ρ1 2 0 ρ1 ρ1 2 1 ≤ − 2 a(1/ρ) ˙ + 2 a(1/ρ) ˙ − ta(t)dt ˙ ≤4 −ta(t)dt, ˙ (3.54) ρ 2ρ 0 0

≥

since by convexity ρ1 ρ1 1 − 2 a(1/ρ) ˙ =( tdt)(−a(1/ρ)) ˙ ≤ −ta(t)dt. ˙ 2ρ 0 0 This proves (ii). To derive the bound from below for Re a ˆ(λ) we use (3.48) with k = 3. ∞ a∞ 1 (−iρt)2 a ˆ(iρ) = − − e−iρt )d¨ (1 − iρt + a(t), iρ (iρ)3 0 2 which implies

∞ 1 Re a ˆ(iρ) = − 3 (ρt − sin ρt)d¨ a(t). ρ 0 Estimating t − sin t as above we obtain ρ1 3 ρ1 1 6 t − 3 (ρt − sin ρt)d¨ a(t) ≥ − d¨ a(t) ρ 0 10 0 3! ρ1 6 1 1 = − [ 3a ¨(1/ρ) − 2 a(1/ρ) ˙ + ta(t)dt]. ˙ 10 6ρ 2ρ 0 On the other hand, ∞ ∞ ∞ 3 7 3 −ρ (ρt − sin ρt)d¨ a(t) ≥ − (ρt − 1)d¨ a(t) − (1 − sin 1)d¨ a(t) 1 10ρ3 ρ1 10ρ3 ρ1 ρ ∞ 3 1 3 1 ≥ a ¨(t)dt + a ¨(1/ρ) ≥ − a(1/ρ) ˙ + a ¨(1/ρ). 2 3 2 10ρ ρ1 10ρ 10ρ 10ρ3 Combining these inequalities yields (iii). 2

4

Subordination

The class of completely positive kernels plays a prominent role in the theory of vector-valued Volterra equations, and appears in applications quite naturally. This class of kernels, its properties and associated creep functions are discussed thoroughly in this section. By means of the principle of subordination it is possible to construct new resolvents from a given one, e.g. from a C0 -semigroup or from a cosine family. The new resolvent can be explicitly represented in terms of the given one, and of the propagation function associated with a completely positive kernel. This representation is particularly useful for the understanding of the regularity and the asymptotic behaviour of the resolvent. 4.1 Bernstein Functions The theorem of Bernstein on completely monotonic functions will play a fundamental role in this section. Before we state this result let us recall the definition of a completely monotonic function on (0, ∞). Definition 4.1 A C ∞ -function f : (0, ∞) → ⺢ is called completely monotonic if (−1)n f (n) (λ) ≥ 0 for all λ > 0, n ∈ ⺞0 . (4.1) The class of completely monotonic functions will be denoted by CM. Bernstein’s theorem gives the characterization of completely monotonic functions as Laplace transforms of positive measures supported on ⺢+ . Bernstein’s Theorem A C ∞ -function f : (0, ∞) → ⺢ is completely monotonic iff there is a nondecreasing function b : ⺢+ → ⺢, such that ∞ f (λ) = e−λt db(t), λ > 0. (4.2) 0

Normalizing b(t) by ‘b(0) = 0 and b(t) left-continuous’, b(t) is uniquely determined by f . Moreover, ∞ n (n) (−1) f (λ) = e−λt tn db(t), λ > 0, n ∈ ⺞0 , (4.3) 0

and (−1)n f (n) (0+ ) =

0

∞

tn db(t),

n ∈ ⺞0 .

Bernstein’s Theorem shows in particular, that every f ∈ CM admits a holomorphic extension to ⺓+ . The class CM is easily seen to be closed under pointwise addition, multiplication and convergence. However, the composition of completely monotonic functions is in general not completely monotonic, for this another class of functions is needed. J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_4, © Springer Basel 1993

90

4. Subordination

91

Definition 4.2 A C ∞ -function h : (α, β) → ⺢ is called absolutely monotonic on (α, β) if h(n) (μ) ≥ 0 for all μ ∈ (α, β), n ∈ ⺞0 . Let AM(α, β) denote this class of functions. Frequently, the following result on composition of functions will be used. Proposition 4.1 Let h ∈ AM(α, β) and f ∈ CM such that f (0+) ≤ β

and f (∞) ≥ α.

(4.4)

Then h ◦ f defined by (h ◦ f )(λ) = h(f (λ)), λ > 0, is completely monotonic. The proof follows by an application of the chain rule and the product rule for differentiation of composite functions. Observe that (h ◦ f )(λ) = h(f (λ)) is welldefined on (0, ∞) by (4.4). Definition 4.3 A C ∞ -function ϕ : (0, ∞) → ⺢ is called a Bernstein function if ϕ(λ) ≥ 0 for λ > 0 and ϕ is completely monotonic. The class of Bernstein functions will be denoted by BF . The class BF is easily seen to be closed under pointwise addition, multiplication with positive numbers and convergence, however, in general not under multiplication. Concerning composition we have Proposition 4.2 Let f ∈ CM and ϕ, ψ ∈ BF. Then (i) f ◦ ϕ is completely monotonic; (ii) ψ ◦ ϕ is a Bernstein function. The proof again follows by application of the chain and the product rule of differentiation. The following respresentation of Bernstein functions will be of central importance in the sequel; it is a consequence of Bernstein’s Theorem. Proposition 4.3 A C ∞ -function ϕ : (0, ∞) → ⺢ is a Bernstein function iff there exist unique constants k0 , k∞ ≥ 0, a unique function k1 ∈ L1loc (⺢+ ), nonnegative and nonincreasing with limt→∞ k1 (t) = 0 such that ϕ(λ) = λ(k0 +

k∞ ˆ + k1 (λ)), λ

λ > 0.

(4.5)

and k∞ = ϕ(0+).

(4.6)

Moreover, ϕ(λ) λ→∞ λ

k0 = lim

Proof: (⇐) Suppose ϕ(λ) is of the form (4.5) with k0 , k∞ ≥ 0, and k1 ∈ L1loc (⺢+ ) nonnegative, nonincreasing. Then ϕ is of class C ∞ on (0, ∞), ϕ(λ) ≥ 0 for all λ > 0; therefore it is sufficient to show that ϕ ∈ CM, i.e. that f (λ) = (d/dλ)(λkˆ1 (λ))

92


is completely monotonic. To verify this, assume first k1 (0+) < ∞. Then k1 ∈ 1 (λ), hence BV (⺢+ ) and λkˆ1 (λ) = dk t t l(t) = − sdk1 (s) = − sdk1 (s) 0

0+

is nondecreasing since k1 is nonincreasing. Therefore 1 ) (λ) = −tdk 1 (λ) = dl(λ) f (λ) = (dk is completely monotonic by Bernstein’s Theorem. If k1 (0+) = ∞, we approximate k1 (t) by k1ε (t) = k1 (t + ε), ε, t > 0, and let ϕε (λ) = λ(k0 +

k∞ + k1ε (λ)), λ

λ > 0.

Then ϕε (λ) is a Bernstein function and ϕε (λ) → ϕ(λ) pointwise as ε → 0+, since ε k1ε (λ) = eλε kˆ1 (λ) − eλ(ε−t) k1 (t)dt → kˆ1 (λ) 0

as ε → 0 + . Hence ϕ(λ) is a Bernstein function. (⇒) Assume that ϕ(λ) is a Bernstein function. Then k∞ = ϕ(0+) exists and is nonnegative; subtracting k∞ from ϕ(λ) we may assume w.l.o.g. k∞ = 0. Since ϕ(λ) is positive and concave, ϕ(λ) ϕ(λ) = inf λ→∞ λ λ>0 λ

k0 = lim

exists and is nonnegative; subtracting k0 λ from ϕ(λ) we may assume w.l.o.g. k0 = 0. Next observe that 1 ϕ(λ) = λ ϕ (tλ)dt, λ > 0; 0

therefore the function ψ(λ) = ϕ(λ)/λ is completely monotonic. Bernstein’s Theorem provides nondecreasing functions k, l, normalized by k(0) = l(0) = 0 and left-continuity, such that ϕ(λ) = λdk(λ) and ϕ (λ) = dl(λ) for λ > 0. Then

ˆl(λ) ˆ ˆ ϕ (λ) (λ2 k(λ)) k(λ) k(λ) = = =2 + kˆ (λ) = 2 − tk(λ), 2 2 λ λ λ λ λ

i.e.

t

tk(t) = 2 0

k(τ )dτ −

t

l(τ )dτ,

t > 0.

0

From this identity and k(0+ ) = k(0) = 0 (which follows from k0 = 0) we obtain 1,1 ˙ k ∈ Wloc (⺢+ ), and with k1 (t) = k(t) tk1 (t) = k(t) − l(t),

for a.a. t > 0.

4. Subordination

93

Extend k1 to all of (0, ∞) by means of this relation; then k1 (t) is nonnegative, left-continuous and tk1 ∈ BVloc (⺢+ ), hence k1 ∈ BV [ε, 1/ε], for every ε > 0. We now show that k1 (t) is nonincreasing. For this purpose, let p ∈ C 1 (0, ∞) be nonnegative and with compact support; then ∞ ∞ ∞ ∞ 0≤ p(t)dl(t) = − p (t)l(t)dt = (tp(t)) k1 (t)dt = − tp(t)dk1 (t), 0

0

0

0

∞

hence 0 q(τ )dk1 (τ ) ≤ 0 for all nonnegative q ∈ C 1 (0, ∞) with compact support, and so also for all such q ∈ C(0, ∞). This implies k1 (t) nonincreasing. Finally, we have lim k1 (t) = lim λkˆ1 (λ) = lim ϕ(λ) = ϕ(0+) = 0, t→∞

λ→0

and from this also (4.6) follows.

λ→0

2

Decomposing k1 (t) = k2 (t) + k3 (t), t > 0, where k2 (t) = max(k1 (t) − k1 (1), 0),

t > 0,

(4.7)

and k3 (t) = min(k1 (t), k1 (1)),

t > 0,

(4.8)

(4.5) becomes 3 (λ), ϕ(λ) = λk0 + λkˆ2 (λ) + k∞ + dk

λ > 0,

(4.9)

since k3 ∈ BV (⺢+ ). As a consequence of (4.9) we obtain Corollary 4.1 Let ϕ(λ) be a Bernstein function. Then ϕ(λ) admits a continuous extension to ⺓+ - again denoted by ϕ(λ) - which is holomorphic in ⺓+ , and satisfies Re ϕ(λ) > 0 for all Re λ > 0. Proof: Since k2 ∈ L1 (⺢+ ) and k3 ∈ BV (⺢+ ), (4.9) shows that ϕ(λ) can be continuously extended to ⺓+ , such that the extension which is again denoted by ϕ(λ) is holomorphic in ⺓+ . To see that Re ϕ(λ) > 0 for Re λ > 0, we assume 1 (λ), Re λ > 0, and since k1 (t) is first k1 (0+) < ∞. Then ϕ(λ) = λk0 + k∞ + dk nonincreasing ∞ ∞ 1 (λ) = k1 (0+) + Re Re dk e−λt dk1 (t) ≥ k1 (0+) + dk1 (t) = 0, 0+

0+

hence Re ϕ(λ) ≥ 0 for all Re λ ≥ 0. If k1 (0+) = ∞, approximate k1 (t) by k1ε (t) = k1 (t + ε), t, ε > 0, as in the proof of Proposition 4.3; then ϕε (λ) = λk0 + k∞ + λk1ε (λ) → ϕ(λ) on ⺓+ as ε → 0+, and therefore Re ϕ(λ) ≥ 0 on ⺓+ . But we cannot have Re ϕ(λ0 ) = 0 for some λ0 ∈ ⺓+ since u(λ) = Re ϕ(λ) is harmonic in ⺓+ . 2

94


In virtue of the representation of Bernstein functions obtained in Proposition 4.3, it is convenient to introduce the following Definition 4.4 A function k : (0, ∞) → ⺢ is called a creep function if k(t) is nonnegative, nondecreasing, and concave. A creep function k(t) has the standard form t

k(t) = k0 + k∞ t +

0

k1 (τ )dτ,

t > 0,

(4.10)

where k0 = k(0+) ≥ 0, k∞ = limt→∞ k(t)/t = inf t>0 = k(t)/t ≥ 0, and k1 (t) = ˙ k(t) − k∞ is nonnegative, nonincreasing, limt→∞ k1 (t) = 0. The class of creep functions will be denoted by CF . Thus Proposition 4.3 states that ϕ ∈ BF iff ϕ(λ) = λdk(λ) for some k ∈ CF . Observe that BF ⊂ CF. The class of Bernstein functions ϕ which are represented as ϕ(λ) = λdk(λ) with some k ∈ BF will be called CBF, the class of complete Bernstein functions. We have the inclusions CBF ⊂ BF ⊂ CF . 4.2 Completely Positive Kernels Let ϕ(λ) be a Bernstein function. Then by Proposition 4.2, the function 1/ϕ(λ) = ( 1z ◦ ϕ)(λ) is completely monotonic, and so by Bernstein’s theorem there exists a = 1/ϕ(λ) for λ > 0. On the unique nondecreasing c ∈ BVloc (⺢+ ) such that dc other hand, by Proposition 4.3, ϕ(λ) = λdk(λ), λ > 0, for some creep function k(t). Thus we have the relation 1 dc(λ) dk(λ) = , λ

λ > 0,

hence by uniqueness of the Laplace transform t t c(t − τ )dk(τ ) = k(t − τ )dc(τ ) = t, 0

(4.11)

t > 0,

(4.12)

0

i.e. c(t) solves the Volterra equation of the first kind (4.12). We have proved the first part of Proposition 4.4 Let k ∈ BVloc (⺢+ ) be a creep function. Then there is a unique nondecreasing function c ∈ BVloc (⺢+ ) such that (4.12) holds. Moreover, we have k∞ > 0 ⇔ c(∞) < ∞,

k∞ c(∞) = 1,

(4.13)

and c(0+)k0 = 0,

c(0+)(k1 (0+) + k∞ ) = 1

if k0 = 0.

(4.13) follows from the identity = lim c(∞) = lim dc(λ) λ→0+

1

ˆ λ→0+ k ∞ + λk0 + λk1 (λ)

=

1 , k∞

(4.14)

4. Subordination

95

while the first part of (4.14) is implied by k0 c(0+) = k0 lim dc(λ) = lim λ→∞

λ→∞

k0 = 0; ˆ λk0 + λk1 (λ) + k∞

for the second part of (4.14) see below. Concerning the regularity of c(t), we have to distinguish three cases. Case 1: k0 > 0. Let c2 denote the solution of the Volterra equation k0 c2 + (k1 + k∞ ) ∗ c2 = k1 + k∞ . Then

1

cˆ =

2

1 (1 − cˆ2 ), λ k0

=

i.e. c(t) = k0−1 t − k0−1 t ∗ c2 (t);

2

λ dk

(4.15)

2,1 this implies that c belongs to Wloc (⺢+ ), and

c1 (t) = c(t) ˙ =

k0−1

−

k0−1

t 0

c2 (τ )dτ,

−c˙1 (t) = k0−1 c2 (t), t > 0,

Therefore we have the relations −c˙1 (0+) = k1 (0+) + k∞ , c1 (0+)2

c1 (0+) = k0−1 ,

(4.16)

since c2 (0+) = lim c2 (t) = lim t→0

t→0

(k1 (t) + k∞ ) (k1 (0+) + k∞ ) = . k0 k0 2

Case 2: k0 = 0, k1 (0+) < ∞. Define k2 (t) = k1 (0+) − k1 (t) for t > 0; then k2 ∈ BV 0 (⺢+ ). We have dc(λ) =

=

1 1 1 = = 2 (λ) λkˆ1 (λ) + k∞ λdk(λ) k1 (0+) + k∞ − dk 2 (λ) 1 dk · ( )n , k1 (0+) + k∞ k1 (0+) + k∞

λ > 0,

n≥0

i.e. c(t) has a jump at t = 0 and (k1 (0+) + k∞ )c(0+) = 1.

(4.17)

If k1 (t) is continuous on (0, ∞), then c(t) has no other jumps, and if k1 (t) is even absolutely continuous on (0, ∞) then c(t) is as well. Case 3: k0 = 0,

k1 (0+) = ∞.

96


Then c(t) cannot have jumps, since otherwise dc(t) ≥ γδ(t − t0 ) for some γ > 0, t0 ≥ 0, hence 1 = (k1 + k∞ ) ∗ dc ≥ γ(k1 + k∞ ) ∗ δ(t − t0 ) = γ(k1 (t − t0 ) + k∞ ) for t > t0 , which yields a contradiction to k1 (0+) = ∞. However, the singular part of c(t) can still be nontrivial, as the example k1 (t) = 1 +

∞

e0 (2−n − t),

k ∞ = k0 = 0

n=1

shows. On the other hand, if c(t) is absolutely continuous on (0, ∞) then c(0+) ˙ = ∞; in fact, otherwise we would have k∞ k∞ 1 = lim λc(λ) ˙ · (kˆ1 (λ) + ) = lim λc(λ) ˙ · lim (kˆ1 (λ) + ) = c(0+) ˙ · 0 = 0. λ→∞ λ→∞ λ→∞ λ λ If k1 (t) is absolutely continuous on (0, ∞), then −tk˙ 1 (t) is integrable near zero; via the representation 2 (λ) = −[dc(λ)] tdc(λ) = −dc [tk˙ 1 (λ)],

λ > 0,

there follows that c(t) is absolutely continuous on (0, ∞). 2

Definition 4.5 Let c ∈ BVloc (⺢+ ). The measure dc is called completely positive if there is a creep function such that (4.12) holds. If c is absolutely continuous on each interval [0, τ ] and dc is completely positive then we call c(t) ˙ a completely positive function. Observe that a completely positive measure dc is Laplace-transformable for λ > 0, dc(λ) is positive, completely monotonic, and satisfies 1 dc(λ) dk(λ) = , λ

λ > 0.

(4.18)

We shall frequently make use of the function s(t; μ) introduced in Section 1.3. s(t; μ) + μ 0

t

s(t − τ ; μ)dc(τ ) = 1,

t, μ > 0.

(4.19)

The next proposition contains several different characterizations of completely positive measures. Proposition 4.5 Suppose c ∈ BVloc (⺢+ ) is Laplace transformable and such that dc(λ) > 0 for all λ > 0. Then the following assertions are equivalent: (i) dc is completely positive; (ii) ϕ(λ) = 1/dc(λ) is a Bernstein Function;

4. Subordination

97

(iii) ψτ (λ) = exp (−τ /dc(λ)) is completely monotonic, for every τ > 0; (iv) ϕμ (λ) = dc(λ)/(1 + μdc(λ)) is completely monotonic, for every μ > 0; (v) s(t, μ) is positive and nonincreasing w.r.t. t > 0, for every μ > 0. Proof: (i) ⇒ (ii). Since dc is completely positive, there is a creep function k(t), such that (4.18) holds. Then ϕ(λ) = 1/dc(λ) = λdk(λ) is a Bernstein function by Proposition 4.3. (ii) ⇒ (iii). If ϕ(λ) = 1/dc(λ) is a Bernstein function, then ψτ (λ) = exp(−τ /dc(λ)) = (exp(−τ z) ◦ ϕ)(λ) is the composition of the completely monotonic function exp(−τ z) and a Bernstein function; Proposition 2 yields ψτ ∈ CM for every τ > 0. (iii) ⇒ (iv). Let ψτ ∈ CM for every τ > 0. We have the identity ϕμ (λ) =

∞ ∞ dc(λ) 1 = = e−τ ϕ(λ) e−μτ dτ = e−μτ ψτ (λ)dτ μ + ϕ(λ) 1 + μdc(λ) 0 0

in view of ϕ(λ) > 0 for λ > 0; so ϕμ ∈ CM for each μ > 0, since CM forms a closed convex cone. (iv) ⇒ (v). Suppose ϕμ ∈ CM for all μ > 0. Since μϕμ ≤ 1 for all λ, μ > 0, −1 λˆ s(λ; μ) = (1 + μdc(λ)) =1−

μdc(λ) = 1 − μϕμ (λ) 1 + μdc(λ)

is a Bernstein function. By Proposition 4.3 there are creep functions k μ (t) such that μ (λ) = λk μ + k μ + λkμ (λ), λ > 0. λˆ s(λ; μ) = λdk ∞ 0 1 By uniqueness of the Laplace transform this yields k0μ = 0 and μ s(t; μ) = k∞ + k1μ (t),

i.e. s(t; μ) is positive and nonincreasing. (v) ⇒ (ii). Let s(t; μ) be positive and nonincreasing. Then μϕ(λ)/(μ + ϕ(λ)) = μλˆ s(λ; μ) is a Bernstein function, and so ϕ(λ) = limμ→∞ μλˆ s(λ; μ) is as well since BF is closed. (ii) ⇒ (i). Let ϕ(λ) = 1/dc(λ) be a Bernstein function. Then by Proposition 4.3 there is a creep function k(t) such that ϕ(λ) = λdk(λ), i.e. (4.18) holds. Hence dc is completely positive. 2

Corollary 4.2 Let dc be a completely positive measure, and let φμ (λ) = λˆ s(λ; μ) = ϕ(λ)/(μ + ϕ(λ)), λ, μ > 0. Then φμ (λ) is a Bernstein function; the corresponding completely positive measure is dcμ = δ0 + μdc, s(0+; μ) = (1 + μc(0+))−1 , and

98


s(∞; μ) = (1 + μc(∞))−1 . Moreover, 1/ϕμ (λ) is a Bernstein function and there is a completely positive measure dr(·; μ) such that dr(λ; μ) = ϕμ (λ). Observe that we have the following relations between s(t; μ) and r(t; μ). s(t; μ) + μr(t; μ) = 1,

t

r(t; μ) = 0

s(t − τ ; μ)dc(τ ), t, μ > 0.

(4.20)

The definition of completely positive measures is rather indirect, and therefore we now give sufficient conditions for measures to be completely positive. For this purpose we need Lemma 4.1 Let b ∈ L1loc (0, ∞) be positive, nonincreasing, and log-convex. Then there is r ∈ L1 (⺢+ ) ∩ C(0, ∞), r(t) ≥ 0 for t > 0, such that r +b ∗ r = b. Moreover, we have 0 ≤ r(t) ≤ b(t), for all t > 0 and

∞

r(t)dt = rˆ(0) = 0

ˆb(0) ≤ 1. 1 + ˆb(0)

If b ∈ CM then r ∈ CM. For a proof of Lemma 4.1 we refer to the monograph of Gripenberg, Londen, Staffans [156] or to the original papers of Friedman [120], Miller [241] and Reuter [290]. 1,1 Consider a creep function c ∈ Wloc (⺢+ ) such that c(t) ˙ is in addition logconvex. With b(t) = μc/(1 ˙ + μc0 ), (4.19) is equivalent to s + b ∗ s = γ, where γ = (1 + μc0 )−1 . Since b(t) satisfies the assumption of Lemma 4.1 we obtain t s(t; μ) = γ(1 − 0 r(τ )dτ ), i.e. s(t; μ) ≥ 0 and s(t; ˙ μ) = −γr(t) ≤ 0 for all t > 0. Therefore s(t; μ) is nonnegative and nonincreasing, and so Proposition 4.5 yields dc completely positive. If in addition t c1 ∈ CM, then r ∈ CM as well, by Lemma 4.1, but also s ∈ CM. Therefore μ 0 s(τ ; μ)dt is a Bernstein function, for each μ > 0. From the identity ∂ 1 ∂ 1 1 dc(λ) sˆ(λ; μ) = =− = −ˆ s(λ; μ)dr(λ; μ) 2 ∂μ λ ∂μ 1 + μdc(λ) λ (1 + μdc(λ)) we obtain ∂ s(t; μ) = −s(t; μ) ∗ dr(t; μ) ≤ 0, ∂μ

for all t, μ > 0,

and so s∞ (t) = limμ→∞ s(t; μ) exists for each t ≥ 0. Since 0 ≤ s(t; μ) ≤ 1 holds, the latter implies s(λ; μ) → sˆ∞ (λ) = 0 as μ → ∞, hence s∞ (t) = 0 a.e. (4.20) implies t therefore μr(t; μ) = dc ∗ μs(t; μ) → 1 pointwise as μ → ∞, hence μ 0 s(τ ; μ)dτ = μr(·; μ) ∗ dk(t) → k(t) for a.a. t > 0, and so k(t) is a Bernstein function, in particular k1 (t) is completely monotonic. We have proved

4. Subordination

99

Proposition 4.6 Let c ∈ BVloc (⺢+ ) be a creep function such that c1 (t) is logconvex. Then dc is completely positive. If in addition c ∈ BF, then k(t) defined by (4.12) is also a Bernstein function. 4.3 The Subordination Principle ∞ Let b ∈ BVloc (R+ ) be nondecreasing, such that 0 db(t)e−λt < ∞ for each λ > 0, and let dc be a completely positive measure. Then f (λ) = db(λ) is completely monotonic, and ϕ(λ) = 1/dc(λ) is a Bernstein function. By Proposition 4.2, f ◦ ϕ is completely monotonic, hence there is a ∈ BVloc (⺢+ ) nondecreasing, such that da(λ) = f (ϕ(λ)) = db(

1 ), dc(λ)

λ > 0.

(4.21)

This is the so-called subordination principle for completely positive measures. We want to use this principle to construct new resolvents Sa (t) from a given resolvent Sb (t). Before we can do this, the subordination principle has to be studied in some detail. ∞ Proposition 4.7 Let b, c ∈ BVloc (⺢+ ) be such that 0 |db(t)|e−βt < ∞, β ≥ 0, and dc be completely positive. Define α ≥ 0 by −1 ( 1 ) α = dc β

1 , α = 0 otherwise. (4.22) β ∞ Then there is a function a ∈ BVloc (⺢+ ) with 0 |da(t)|e−αt < ∞ such that (4.21) holds for all λ > α. Moreover, (a) b nondecreasing ⇒ a nondecreasing; (b) db completely positive ⇒ da completely positive; (c) b, c Bernstein functions ⇒ a Bernstein function; 1,1 1,1 (d) b, c ∈ Wloc (⺢+ ) ⇒ a ∈ Wloc (⺢+ ). if c(∞) >

Proof: Let b and c satisfy the assumptions of Proposition 4.7; by means of the Jordan decomposition of b we may w.l.o.g. assume that b(t) is nondecreasing. From the definition of α in (4.22) it is clear that g(λ) = db(1/ dc(λ)) is well-defined for λ ≥ α and g(λ) ≤ db(1/dc(α)) = db(β) < ∞ on [α, ∞). It follows from Proposition 4.2 that g(λ) is completely monotonic on (α, ∞), hence Bernstein’s theorem yields ∞ a nondecreasing function a(t) such that in addition 0 da(t)e−αt = g(α) < ∞. This proves the statement as well as (a); (b) follows from Proposition 4.2 (ii) and from Proposition 4.5. To prove (c) let b, c ∈ BF. Then by Bernstein’s theorem, there is a function ∞ β ∈ BVloc (⺢+ ) with 1 dβ(s)/s < ∞ such that ∞ 1 db(z) = b0 + dβ(μ), z > 0; z + μ 0 hence we have 1 1 b0 a ˆ(λ) = db( )= + λ λ dc(λ)

∞ 0

c(λ) dβ(μ), 1 + μdc(λ)

λ > 0.

100


This yields the representation

a(t) = b0 +

∞

r(t; μ)dβ(μ),

t > 0,

0

where r(t; μ) denotes as usual the solution of r+μdc∗r = c. This integral converges absolutely for each t > 0, since μr(t; μ) ≤ 1 holds. By Lemma 4.1, c ∈ BF implies r(·; μ) ∈ BF for each μ > 0, and so a(t) is again a Bernstein function. It remains to prove (d). For this purpose we let ∞ BVβ (⺢+ ) = {b ∈ BVloc (⺢+ ) : |b|β = |db(t)|e−βt < ∞}; 0

BVβ (⺢+ ) equipped with the norm |b|β is a Banach space. We have shown that the linear operator C : BVβ (⺢+ ) → BVα (⺢+ ) defined by Cb = a, with a given by (4.21) is well-defined. (4.21) also implies the closedness of C, hence C is bounded by the closed graph theorem. Since 1,1 ACβ (⺢+ ) = BVβ (⺢+ ) ∩ Wloc (⺢+ )

is a closed subspace of BVβ (⺢+ ), it is therefore sufficient to show that C maps a subset E ⊂ ACβ (⺢+ ) with spanE = ACβ (⺢+ ) into ACα (⺢+ ). So let c ∈ ACloc (⺢+ ) be such that dc is completely positive and set E = {e−μt : μ > −β}. Then spanE = ACβ (⺢+ ) and for b ∈ E we have c(λ) ˙ , 1 + μc(λ) ˙

da(λ) =

λ ≥ α.

Since 1 + μc(λ) ˙ = 0 for Re λ ≥ α, the Paley-Wiener lemma implies a ∈ ACα (⺢+ ). This completes the proof. 2

Consider now the Volterra equation t u(t) = f (t) + b(t − τ )Au(τ )dτ,

t > 0,

(4.23)

0

where A is closed linear and densely defined, the kernel b ∈ L1loc (⺢+ ) is such ∞ that 0 |b(t)|e−βt dt < ∞, and let c(t) be a completely positive function. Define 1 a ∈ Lloc (⺢+ ) by 1 a ˆ(λ) = ˆb( ), λ ≥ α, (4.24) cˆ(λ) according to Proposition 4.7, where α is given by ∞ 1 1 α = cˆ−1 ( ) if c(t)dt > , α = 0 otherwise. (4.25) β β 0 The Volterra equation v(t) = g(t) + 0

t

a(t − τ )Av(τ )dτ,

t > 0,

(4.26)

4. Subordination

101

is then called subordinate to (4.23) via c(t). We now prove that (4.26) admits a resolvent whenever (4.23) does; this is the general subordination principle for resolvents. Theorem 4.1 Let A be a closed ∞ linear and densely defined operator in X, and let b, c ∈ L1loc (⺢+ ) such that 0 |b(t)|e−βt dt < ∞ for some β ∈ ⺢. Assume (i) c(t) is a completely positive function; (ii) there is a resolvent Sb (t) of (4.23) of type (M, ωb ), ωb ≥ 0. // Then there is a resolvent Sa (t) for (4.26). Moreover, let ωa be defined by ∞ 1 1 ωa = cˆ−1 ( ) if c(t)dt > , ωa = 0 otherwise; (4.27) ωb ωb 0 then there is a constant Ma ≥ 1 such that |Sa (t)| ≤ Ma eωa t , |Sa (t)| ≤ Mε eεt ,

t > 0, if ωb = 0, or ωb cˆ(0) = 1, t > 0, if ωb > 0 and ωb cˆ(0) = 1,

(4.28)

where ε > 0 is arbitrary. Proof: Let A, b, c satisfy the assumptions of Theorem 4.1 and let a(t) be defined by (4.24), according to Proposition 4.7. Then ∞ 1 Sˆb (z) = e−zτ Sb (τ )dτ = (I − ˆb(z)A)−1 , z > ωb , z 0 and so we obtain H(λ) =

1 1 1 (I − a ˆ(λ)A)−1 = · cˆ(λ)(I − ˆb( )A)−1 λ λˆ c(λ) cˆ(λ) ∞ 1 = e−τ /ˆc(λ) Sb (τ )dτ λˆ c(λ) 0

for λ sufficiently large. With h(λ; τ ) =

1 e−τ /ˆc(λ) , λˆ c(λ)

λ > 0, τ ≥ 0,

this yields the representation H(λ) = 0

∞

h(λ; τ )Sb (τ )dτ,

λ > ωa .

Proposition 4.5 shows that h(λ; τ ) is completely monotonic on (0, ∞) for each τ > 0, hence with Lnλ = (−1)n n!−1 (d/dλ)n we obtain ∞ ∞ |Lnλ H(λ)| ≤ Lnλ h(λ; τ )|Sb (τ )|dτ ≤ M Lnλ h(λ; τ )eωb τ dτ 0

0

102


=

M Lnλ

0

∞

h(λ; τ )eωb τ dτ = M Lnλ [

1 ] = M Lnλ [ˆ s(λ; −ωb )], λ(1 − ωb cˆ(λ))

λ > ωa .

Suppose for a moment that s(t; −ωb ) ≤ meωt ,

t > 0,

(4.29)

is satisfied. Since s(t; −ωb ) ≥ 0 we obtain by (4.29) 0 ≤ Lnλ [ˆ s(λ; −ωb )] ≤ hence |Lnλ H(λ)| ≤

m , (λ − ω)n+1

mM , (λ − ω)n+1

λ > ω, n ∈ ⺞0 ,

(4.30)

λ > ω, n ∈ ⺞0 .

The generation theorem, Theorem 1.3, then applies and therefore equation (4.26) admits a resolvent of type (mM, ω). To verify (4.29) we have to consider several cases. a) ωb = 0 This is the simplest case, since then s(t; −ωb ) ≡ 1, i.e. (4.29) holds with m = 1 and ω = ωa = 0. b) ωb cˆ(0) < 1 , ωb > 0. In this case c ∈ L1 (⺢+ ) and 1−ωb cˆ(λ) = 0 for all Re λ ≥ 0 since c(t) is nonnegative; the Paley-Wiener Lemma then yields r(·; −ωb ) ∈ L1 (⺢+ ), where rˆ(λ; −ωb ) =

cˆ(λ) , 1 − ωb cˆ(λ)

Re λ ≥ 0.

From this we obtain s(t; −ωb ) = 1 + ωb

t 0

r(τ ; −ωb )dτ,

t > 0,

bounded, i.e. (4.29) holds with ω = ωa = 0 and m = (1 − ωb cˆ(0))−1 ; observe that r(t; −ωb ) ≥ 0, i.e. s(t; −ωb ) is nonnegative and nondecreasing. c) ωb cˆ(0) > 1, ωb > 0. In this case ωa > 0, and λ = ωa is a simple pole of sˆ(λ; −ωb ), since this function is completely monotonic, and analytic in ⺓+ , except for poles, which are the zeros of 1 − ωb cˆ(λ). Since by positivity of c(t) no other poles are on the line Reλ = ωa , and s(t; −ωb ) is nonnegative, nondecreasing, we may apply Ikehara’s Theorem (cp. Widder [339], Thm V. 17]) to the result lim s(t; −ωb )e−ωa t = −

t→∞

cˆ(ωa ) . ωa cˆ (ωa )

This shows that (4.29) holds with ω = ωa and some m ≥ 1. d) ωb cˆ(0) = 1, ωb > 0. Then c ∈ L1 (⺢+ ), hence sˆ(λ; −ωb ) is analytic in ⺓+ , continuous in ⺓+ \ {0}, and

4. Subordination

103

λ = 0 is a singularity of higher order, therefore s(t; −ωb ) cannot be bounded. However, by the Paley-Wiener lemma, r(t; −ωb )e−ωt is integrable for every ω > 0. We conclude that (4.29) holds for every ω > 0. 2

It should be noted that there is a result similar to Theorem 4.1 for the integral resolvents Rb (t) and Ra (t) defined in Section 1.6; see Section 4.7. 4.4 Equations with Completely Positive Kernels Suppose A is the generator of a C0 -semigroup of type (M, ω) in X, and let c(t) be a completely positive function. Then (4.23) with b(t) ≡ 1 admits a resolvent of type (M, ω), and we are in position to apply Theorem 4.1. For the kernel a(t) defined by (4.24) we obtain with ˆb(z) = 1/z a ˆ(λ) = ˆb(1/ˆ c(λ)) = cˆ(λ),

λ > 0,

i.e. (4.26) becomes

t

v(t) = g(t) + 0

c(t − τ )Av(τ )dτ,

t > 0.

(4.31)

We have proved Theorem 4.2 Let A be the generator a C0 -semigroup of type (M, ω) in X, and let c ∈ L1loc (⺢+ ) be a completely positive function. Then (4.31) admits a resolvent S(t) in X which is exponentially bounded. Moreover, if ωc ≥ 0 is defined by ∞ 1 1 ωc = cˆ−1 ( ) if c(t)dt > , ωc = 0 otherwise , (4.32) ω ω 0 then S(t) is of type (Mc , ωc ) if ω = 0 or ωˆ c(0) = 1, and of type (Mε , ε) for every ε > 0 if ω > 0, ωˆ c(0) = 1. Next suppose A generates a cosine family T (t) in X such that |T (t)| ≤ M eωt for t > 0 holds, and let c(t) be again completely positive. Then (4.23) with b(t) = t admits a resolvent of type (M, ω). For the kernel a(t) defined by (4.24) we obtain in this case 1 1 a ˆ(λ) = ˆb( )= = cˆ(λ)2 , λ > 0, cˆ(λ) cˆ(λ)−2 since ˆb(z) = 1/z 2 ; this implies a(t) = (c ∗ c)(t). By Theorem 4.1, (4.26) admits an exponential bounded resolvent. In practice, however, (4.26) is given, and so there remains the problem to determine which kernels a(t) are squares (with convolution as the multiplication) of completely positive functions; this amounts to showing that ϕ(λ) = 1/ a ˆ(λ) is a Bernstein function. A necessary condition for this obviously is a ˆ(λ) > 0 for λ > 0, but of course it is far from sufficient. A first sufficient condition is prevailed by Proposition 4.2; if a(t) itself is completely positive, then 1/ˆ a(λ) is a Bernstein function by Proposition 4.5, hence

104


√ 1/ a ˆ(λ) is again a Bernstein function by Proposition 4.2, since λ belongs to this class. From the applications point of view, creep functions a(t) are important; see Section 5. This is in general not enough for ϕ(λ) = 1/ a ˆ(λ) to belong to the class BF, however, we have the following. Lemma 4.2 Suppose a(t) ≡ 0 is a creep function with a1 (t) log-convex. Then ϕ(λ) = (ˆ a(λ))−1/2 is a Bernstein function, and there is a completely positive function c ∈ L1loc (⺢+ ) such that the factorization a = c ∗ c holds. In addition, if a ∈ BF then c ∈ CM. Proof: Let a(t) be a creep function with a1 (t) log-convex and let s ≥ 0. By Proposition 4.6, das = da + sdt is a completely positive measure, hence ϕs (λ) = s (λ) is a Bernstein function for each s ≥ 0. The complex variable formula 1/da 1 ∞ 1 dr 1 √ = ·√ , z∈ / ⺢− z π 0 z+r r then yields the representation ϕ(λ) =

1 π

∞ 0

1 dr √ , a ˆ(λ) + r r

λ > 0,

for ϕ(λ) = (ˆ a(λ))−1/2 . The change of variables r = s/λ2 gives 1 ∞ ds ϕ(λ) = ϕs (λ) √ , λ > 0; π 0 s

(4.33)

observe that the integral (4.33) for ϕ(λ) is absolutely convergent, for each λ > 0, as are those for its derivatives. Since ϕs (λ) is a Bernstein function for every s ≥ 0, and the class BF is a closed convex cone, ϕ(λ) is also a Bernstein function. This proves the first part of the Lemma. Thus by Proposition 4.5, there is a completely positive measure dc such that a = dc ∗ dc. The measure dc, however, cannot have a singular part since a(t) is absolutely continuous on (0, ∞). The last part of the lemma is a special case of Lemma 4.3 below. 2

Summarizing we have the following result. Theorem 4.3 Let A generate a cosine family T (t) in X such that |T (t)| ≤ M eωt for t > 0 holds, and assume any of the following. (i) a ∈ L1loc (⺢+ ) is completely positive; (ii) a(t) is a creep function with a1 (t) log-convex; (iii) a = c ∗ c with some completely positive c ∈ L1loc (⺢+ ). Then (4.36) admits a resolvent S(t) with growth bound ωc determined by ωc = a ˆ−1/2 (ω −1 ) if a ˆ(0) > ω −2 ,

ωc = 0 otherwise.

(4.34)

In passing we mention one particular case of Theorem 4.3 which is well-known.

4. Subordination

105

Corollary 4.3 Let A generate a cosine √ family of type (M, ω), ω ≥ 0. Then A generates a C0 -semigroup of type (M1 , ω), for some M1 ≥ 1. In fact, choose a(t) ≡ 1 in Theorem 4.3 (ii) to see this. As another application of Theorem 4.1 consider b ∈ L1loc (⺢+ ) which is 1-regular and such that (4.23) is parabolic. Then by Theorem 3.1, (4.23) admits a bounded resolvent. If c(t) is a completely positive function and a(t) is defined by (4.24), then (4.26) is again parabolic, and Theorem 4.1 yields a bounded resolvent S(t). Corollary 4.4 Suppose b ∈ L1loc (⺢+ ) is 1-regular and such that (4.23) is parabolic; let c(t) be a completely positive function. Then (4.26), with a(t) defined by (4.24), is parabolic and admits a resolvent S(t) of type (M, 0), for some M ≥ 1. Observe that in the situation of Corollary 4.4, although (4.26) is parabolic, Theorem 3.1 does not apply since it is not clear whether a(t) is 1-regular again. The second part of Lemma 4.2 is a special case of the following composition rule involving three Bernstein functions. It will be very useful in applications. Lemma 4.3 Suppose a, b, c are Bernstein functions. Then there is a Bernstein function e such that ˆ(λ) ˆ a eˆ(λ) = a ˆ(λ)dc( ), λ > 0. ˆb(λ) Moreover, e0 = a0 (c0 + c1 (a0 /b0 )) for a0 > 0, e0 = 0 otherwise. Proof: (a) We first prove the lemma for the case c(t) = 1 − e−t , i.e. we show that a ˆˆb/(ˆ a + ˆb) is a complete Bernstein function. Let k(t) and l(t) denote the solutions of k ∗ da ≡ t ≡ l ∗ db; by Proposition 4.6, k, l ∈ BF, hence also k + l ∈ BF and so there is m ∈ BF such that m ∗ (dk + dl) ≡ t, by Proposition 4.6 again. The Laplace transform of m is given by m(λ) ˆ =

1 λ (dk(λ) + dl(λ)) 2

=(

1 1 −1 a ˆ(λ)ˆb(λ) , + ) = a ˆ(λ) ˆb(λ) a ˆ(λ) + ˆb(λ)

λ > 0.

(b) Consider now the general case. Since c(t) is a Bernstein function by assumption, there is a function β ∈ BVloc (⺢+ ) such that ∞ c1 (t) = e−st dβ(s), t > 0, 0

∞ a(λ)/ˆb(λ)), λ > 0; then we have the and 1 dβ(s)/s < ∞. Let f (λ) = a ˆ(λ)dc(ˆ representation f (λ) = c0 a ˆ(λ) + c∞ˆb(λ) +

0

∞

a ˆ(λ)ˆb(λ) dβ(s), a ˆ(λ) + sˆb(λ)

λ > 0.

106


Let ms ∈ BF be defined by m s = sˆ aˆb/(ˆ a + sˆb), s > 0; then for each t > 0, ms (t) is nondecreasing w.r.t. s > 0 and lims→∞ ms (t) = a(t), as well as lims→0+ ms (t)/s = b(t). Therefore, e(t) defined by ∞ e(t) = c0 a(t) + c∞ b(t) + ms (t)dβ(s)/s, t > 0 0

exists, the integral being absolutely convergent. Since a, b, ms ∈ BF there follows e ∈ BF, for BF is a closed convex cone. Obviously, eˆ(λ) = f (λ) and so the assertion of the lemma follows. 2

√ √ The choice c(t) = (2/ π) t and b(t) = t yields 1 ˆ(λ) a a ˆ(λ)dc( )=a ˆ(λ)[λ2 a ˆ(λ)]− 2 = ˆb(λ)

a ˆ(λ)/λ,

therefore by Lemma 4.3 the second part of Lemma 4.2 follows. As another applica1−αˆ tion of Lemma 4.3, a ˆ(λ) b(λ)α is the Laplace transform of a Bernstein function, whenever a, b ∈ BF and α ∈ [0, 1]. 4.5 Propagation Functions Let dc be a completely positive measure, ϕ(λ) = 1/dc(λ) its associated Bernstein function, and let k ∈ BVloc (⺢+ ) be the creep function such that ϕ(λ) = λdk(λ), λ > 0. k(t) then has the unique representation (4.10) and for brevity we let κ, ω, α ∈ ⺢+ in the sequel be defined by κ = k0 , ω = k∞ , α = k1 (0+) + k∞ .

(4.35)

By Proposition 4.5, the functions ψτ (λ) = exp(−τ ϕ(λ)) are completely monotonic with respect to λ > 0, for each fixed τ ≥ 0, and bounded by e−τ ϕ(0+) = e−τ ω . From Bernstein’s theorem it follows that there are unique nondecreasing functions w(·; τ ) ∈ BV (⺢+ ), normalized by w(0; τ ) = 0 and left-continuity, such that w(λ; τ) =

ψτ (λ) , λ

λ > 0, τ ∈ ⺢+ .

(4.36)

Obviously, w(·; τ ) enjoys the semigroup property 0

t

w(t − s; τ1 )dw(s; τ2 ) = w(t; τ1 + τ2 ), w(t; 0) = e0 (t),

t ≥ 0,

t, τ1 , τ2 ≥ 0, (4.37)

where e0 (t) denotes the Heaviside function. In the sequel w(t; τ ) will be called the propagation function associated with the completely positive measure dc (or with the Bernstein function ϕ). The properties of w(t; τ ) will be studied in detail below.

4. Subordination

107

The space BV (⺢+ ) becomes a commutative Banach algebra with unit e0 (t), when the multiplication • in BV (⺢+ ) is defined by (a • b)(t) = (a ∗ db)(t) = (b ∗ da)(t), and the norm in BV (⺢+ ) is as usual ||a|| = Var

a|∞ 0

∞

= 0

t ≥ 0,

|da(t)|;

(4.38)

(4.39)

see e.g. Gelfand, Raikov and Shilov [128]. By BV + (⺢+ ) we denote the closed convex cone of nondecreasing functions in BV (⺢+ ); this space then becomes a Banach lattice w.r.t. the ordering generated by BV + (⺢+ ). By virtue of the pairing ∞ < p, a > = p(t)da(t), p ∈ C0 (⺢+ ), a ∈ BV (⺢+ ), (4.40) 0

BV (⺢+ ) can be identified with the dual of C0 (⺢+ ), and therefore carries the weak*-topology of C0 (⺢+ ). Consider the map w : ⺢+ → BV (⺢+ ) defined by w(τ )(t) = w(t; τ ),

t, τ ≥ 0;

(4.41)

the identity (4.37) then shows that the family {w(τ )}τ ≥0 forms a semigroup in BV (⺢+ ) which moreover is positive and satisfies ||w(τ )|| ≤ e−τ ω , τ ≥ 0, i.e. is of type (1, −ω). Besides, w is weak*-continuous; in fact, with eλ (t) = e−λt , t > 0, we have < eλ , w(τ ) >= ψτ (λ), λ > 0, τ ∈ ⺢+ , (4.42) hence the functions < eλ , w(·) > are continuous on ⺢+ ; since ||w(τ )|| ≤ 1 and span {eλ : λ > 0} is dense in C0 (⺢+ ) it follows that w is weak*-continuous. From the definition of ψτ (λ) we obtain ∂ w(τ dc ˆ ) + w(τ ˆ ) = 0, ∂τ w(0) ˆ = eˆ0 , which formally is equivalent to the Volterra equation t ∂ w(t; τ ) + w(t − s; τ )dc(s) = 0, 0 ∂τ w(t; 0) = 1, t > 0; w(0; τ ) = 0,

(4.43)

t, τ ≥ 0, τ ≥ 0.

(4.44)

Definition 4.6 A function w : ⺢+ → BV (⺢+ ) is called a solution of (4.44) if for each p ∈ Cc (⺢+ ), the function < p, w(·) > is continuous on ⺢+ , < p, c • w(·) > is continuously differentiable on ⺢+ , and d < p, c • w(τ ) > + < p, w(τ ) > = 0, w(0) = e0 , dτ is satisfied.

τ ≥ 0,

(4.45)

108


We now show that completely positive measures can be characterized in terms of (4.44). Proposition 4.8 Let c ∈ BVloc (⺢+ ) be of subexponential growth. Then the measure dc is completely positive iff (4.44) admits a solution w : ⺢+ → BV + (⺢+ ) such that |w(τ )| ≤ 1 for τ ≥ 0. If this is the case, the solution is unique. Proof: (⇒) Suppose dc is completely positive, and let c ∈ BV (⺢+ ) first. Then (4.43) is equivalent to τ < eλ , c • w(τ ) > + < eλ , w(σ) > dσ = < eλ , c > (4.46) 0

for all λ > 0. Since span{eλ : λ > 0} is dense in C0 (⺢+ ), and |w(τ )| ≤ 1, (4.46) yields τ < p, w(σ) > dσ = < p, c > (4.47) < p, c • w(τ ) > + 0

for all τ ≥ 0 and p ∈ C0 (⺢+ ). From this it follows that w(τ ) is a solution of (4.44) in the sense of Definition 4.6. If c ∈ BV (⺢+ ) we have to be a little more careful. Observe that (4.43) is equivalent to τ < eλ , [c • w(τ )]ε > + < eλ , [w(σ)]ε > dσ = < eλ , cε > (4.48) 0

for all λ, ε > 0, where we used the notation aε (t) =

t

e−εs da(s),

0

t ≥ 0;

by assumption cε ∈ BV (⺢+ ). Then (4.48) is equivalent to τ < p, [c • w(τ )]ε > + < p, [w(σ)]ε > dσ = < p, cε >

(4.49)

0

for all p ∈ Cc (⺢+ ), and letting ε → 0 we obtain τ < p, c • w(τ ) > + < p, w(τ ) > dτ = < p, c > . 0

This implies that w(τ ) is a solution of (4.44) in the sense of Definition 4.6. (⇐) Conversely, let w : ⺢+ → BV + (⺢+ ) be a solution of (4.44) with |w(τ )| ≤ 1, ∞ −εt τ ≥ 0, and let 0 e |dc(t)| < ∞ for all ε > 0. Passing backwards from (4.45) to (4.43) we obtain · w(τ dc )+

τ

0

w(σ)dσ = c,

λ, τ > 0.

4. Subordination

109

Let γ = dc(λ) and ϕ(τ ) = w(τ ), λ > 0 fixed. Then we obtain γϕ (τ ) + ϕ(τ ) = 0,

τ ≥ 0, ϕ(0) = 1;

therefore, γ = 0 since ϕ is of class C 1 , and ϕ(τ ) = e−τ /γ , τ ≥ 0. Since ||w(τ )|| ≤ 1, τ ≥ 0, there follows γ > 0. Thus we have γ = dc(λ) > 0 for all λ > 0 and w(λ; ˆ τ ) = exp(−τ /dc(λ)) for all τ > 0, λ > 0. Since w(τ ) ∈ BV + (⺢+ ), w(λ; ˆ τ ) is completely monotonic w.r.t. λ > 0, for each τ ≥ 0. Proposition 4.5 now implies that dc is completely positive. Uniqueness of the solution follows from dc(λ) >0 for all λ > 0. 2

Some properties of the propagation function are collected in Proposition 4.9 Let dc be completely positive, and let w denote its associated propagation function. Then we have (i) w(·; ·) is Borel measurable on ⺢+ × ⺢+ ; (ii) w(·; τ ) is nondecreasing and left-continuous on ⺢+ , and w(0; τ ) = 0, w(∞; τ ) = e−ωτ ,

for all τ > 0.

(iii) w(t; ·) is nonincreasing and right-continuous on ⺢+ , and w(t; 0) = w(t; 0+) = 1, w(t; ∞) = 0,

for all t > 0;

(iv) w(t; τ ) = 0 iff t ≤ κτ, and w(κτ ; τ −) = w(κτ +; τ ) = e−ατ , τ ≥ 0. (v) s(t; μ) defined by (4.19) is represented as ∞ s(t; μ) = − e−μτ dτ w(t; τ ), t, μ > 0. 0

Proof: (i) follows from the Post-Widder inversion formula for the Laplace transform by virtue of w(t; τ ) = lim w(t − m→∞

1 n n ; τ ) = lim lim Lnλ w( ˆ ; τ )( )n+1 m→∞ n→∞ m t − 1/m t − 1/m

which holds for every t, τ > 0, τ ≥ 0; cp. Widder [339], Corollary VII. 6a.3. (ii) is implied by Bernstein’s theorem and by the normalization of functions of bounded variation; observe that limt→∞ w(t; τ ) = limλ→0 ψτ (λ) = e−ωτ , τ ≥ 0. (iii) The semigroup property implies t w(t; τ1 + τ2 ) = w(t − s; τ1 )dw(s; τ2 ) ≤ w(∞; τ1 )w(t; τ2 ) ≤ w(t; τ2 ) 0

for all t, τ1 , τ2 ≥ 0, since w(·; τ ) is nonnegative and nondecreasing; therefore w(t; ·) is nonincreasing for fixed t ≥ 0. Moreover, w(t; 0) = 1 is obvious from ψ0 (λ) ≡ 1, and with w(t; ∞) = limτ →∞ w(t; τ ) ≤ w(t; τ ) we obtain 0 ≤ w(λ; ˆ ∞) ≤ w(λ; ˆ τ) =

ψτ (λ) →0 λ

as τ → ∞, λ > 0,

110


hence w(t; ∞) ≡ 0 by uniqueness of the Laplace transform. Finally, observe that ∗ w(·; τ + h)w(·; τ ) as h → 0+ implies w(t; τ + h) → w(t; τ ) as h → 0 for each t ≥ 0, since w(·; τ ) is left-continuous; therefore w(t; ·) is right-continuous. (iv) Assume first κ = 0; we show that w(t; τ ) > 0 for all t > 0, τ ≥ 0. In fact, if w(t; τ ) = 0 for 0 < t ≤ t0 , and some fixed τ > 0, then ∞ 1 −τ ϕ(λ) 1 1 e = w(t; τ )e−λt dt ≤ e−λt0 w(∞; τ ) = e−λt0 −ωτ , λ λ λ 0 hence τ ϕ(λ) ≥ λt0 + ωτ , which implies kˆ1 (λ) ≥ t0 /τ > 0 for all λ > 0; this yields a contradiction to kˆ1 (λ) → 0 as λ → ∞. Moreover, with λkˆ1 (λ) → k1 (0+) as λ → ∞ we obtain w(0+; τ )

=

lim w(t; τ ) = lim λw(λ; ˆ τ ) = exp(−τ k∞ − τ lim λkˆ1 (λ))

t→0+

λ→∞

λ→∞

= exp(−(k∞ + k1 (0+))τ ) = e−ατ ,

τ ≥ 0.

If κ = 0, then w(λ; ˆ τ ) = e−κλτ w ˆ1 (λ; τ ), where w1 (t; τ ) denotes the propagation function associated with the Bernstein function ϕ1 (λ) = ϕ(λ) − κλ. Hence w(t; τ ) = w1 (t − κτ ; τ )e0 (t − κτ ),

t, τ ≥ 0,

and (iv) is proved. (v) follows with an integration by parts from the identity (

∞

e−μτ w(t, τ )dτ )∧ (λ) =

0

1 λ

∞ 0

e−μτ ψτ (λ)dτ =

1 dc(λ) . λ 1 + μdc(λ) 2

Compare the representation of s(t; μ) in Proposition 4.9 (v) with that in Section 3.6. 4.6 Structure of Subordinated Resolvents Let c ∈ L1loc (⺢+ ) be a completely positive function, b ∈ L1loc (⺢+ ) Laplace transformable, and let A be a closed linear densely defined operator in X such that (4.23) admits a resolvent Sb (t) of type (M, ωb ) in X. According to Theorem 4.1, there is an exponentially bounded resolvent Sa (t) for (4.26), where a ∈ L1loc (⺢+ ) is given by (4.24). Since − and Sâ (λ) =

1 λˆ c(λ)

∂ e−τ /ˆc(λ) e−τ /ˆc(λ) = , ∂τ λ λˆ c(λ)

∞ 0

Sb (τ )e

−τ /ˆ c(λ)

τ, λ > 0,

dτ = −

∞ 0

Sb (τ )

∂ w(λ; ˆ τ )dτ, ∂τ

with w as in Section 4.5, we obtain the following representation of Sa (t) in terms of Sb (t) and the propagation function w(t; τ ) associated with c.

4. Subordination

111

Corollary 4.5 Let the assumptions of Theorem 4.1 be satisfied, and let w(t; τ ) be the propagation function associated with c ∈ L1loc (⺢+ ). Then the resolvents Sa (t) and Sb (t) of (4.26) and (4.23), respectively, are related by ∞ Sa (t) = − Sb (τ )dτ w(t; τ ), t > 0; (4.50) 0

(4.50) holds in the strong sense. Since w(t; ·) is nonincreasing, and since w(t; 0) = 1, w(t; ∞) = 0 holds, (4.51) shows that the resolvent Sa (t) of (4.26) is the strong limit of convex combinations of Sb (τ ). We note a simple consequence of this observation. Corollary 4.6 Under the assumptions of Corollary 4.5, let D ⊂ X be a closed convex set, such that Sb (t)D ⊂ D for all t ≥ 0. Then Sa (t)D ⊂ D for all t ≥ 0. In particular, if Sb (t) is positive with respect to some closed convex cone K, then Sa (t) is positive as well. The structure of Sa (t) will now be studied in more detail. For this purpose, we need the following regularity result for the propagation function. Proposition 4.10 Let dc be completely positive, let w(t; τ ) denote the associated propagation function, and let κ, ω, α be as in (4.35). Then w(t; τ ) = e−ωτ w0 (t − κτ ; τ ) + e−ατ e0 (t − κτ ),

t, τ > 0,

(4.51)

where w0 (·; τ ) is nondecreasing, left-continuous, w0 (t; ·) is right-continuous, nonincreasing, and w0 (t; τ ) > 0 iff t > 0; w0 (0+; τ ) = 0. Moreover, (i) k1 ∈ C(0, ∞) implies w0 ∈ C(⺢+ × ⺢+ \{(0, 0)}); 1,1 1,1 (ii) k1 ∈ Wloc (0, ∞) implies w0 ∈ Wloc ((0, ∞) × (0, ∞)). Proof: Decomposition (4.51) and the properties of w0 follows from Proposition 4.9; it remains to prove (i) and (ii). Observe first that we may assume κ = ω = 0. In fact, κ > 0 only yields a shift of amount κτ for w in t-direction, and ω > 0 leads to multiplication with e−ωτ ; these operations do not change the regularity of w and w0 . If α = ∞, then w = w0 ; if α < ∞ then w ˆ0 (λ; τ ) =

1 −τ ϕ(λ) (e − e−ατ ), λ

(4.52)

hence in both cases 1 t w0 (λ; τ ) = −w ˆ0 (λ; τ ) = w ˆ0 (λ; τ ) + τ ϕ (λ)w(λ; ˆ τ ), λ

λ, τ > 0.

112


As in the proof of Proposition 4.3, we have the relation ϕ (λ) = (λkˆ1 (λ)) = dl(λ), t hence with l(t) =− 0 sdk1 (s) tw0 (t; τ ) =

t 0

w0 (s; τ )ds + τ

t 0

l(t − s)dw(s; τ ),

t, τ ≥ 0.

(4.53)

t With l(t) = 0 k1 (s)ds − tk1 (t), we have l(0) = l(0+) = 0 and l ∈ C(⺢+ ) in case k1 ∈ C(0, ∞). Then (4.53) implies w0 (·; τ ) continuous on (0, ∞), uniformly for bounded τ ; since w(0+; τ ) = e−ατ by Proposition 4.9 (iv), assertion (i) follows. 1,1 ˙ If k1 ∈ Wloc (0, ∞) then l is absolutely continuous, l(t) = −tk˙ 1 (t) a.e., and so w0 (·; τ ) is absolutely continuous for each τ ≥ 0, w˙ 0 (·; τ ) ∈ L1 (⺢+ ) for each τ ≥ 0, |w˙ 0 |1 = w0 (∞; τ ) ≤ 1. To prove that w˙ 0 (·; ·) is product-measurable, let ϕε ∈ C0∞ (⺢) be mollifiers. Then ϕε ∗ w˙ 0 (·; τ ) → w˙ 0 (·; τ ) in L1 (⺢) for each fixed τ > 0, and from ϕε ∗ w˙ 0 (·; τ ) = ϕ˙ ε ∗ w0 (·; τ ) and continuity of w0 , the map τ → ϕε ∗ w˙ 0 (·; τ ) is continuous from (0, ∞) to L1 (⺢+ ). This implies that w˙ 0 : (0, ∞) → L1 (⺢+ ) is strongly measurable, hence w˙ 0 (·; ·) is measurable on ⺢+ × ⺢+ . Finally, −

∂ ϕ(λ) −τ /ϕ(λ) ϕ(λ) − α −ατ w ˆ0 (λ; τ ) = (e e − e−ατ ) + , ∂τ λ λ

τ, λ > 0,

i.e. for each fixed τ > 0 we have −

∂ w0 (t; τ ) = k1 ∗ w˙ 0 (t; τ ) + e−ατ (k1 (t) − α), ∂τ

1,1 This implies w0 ∈ Wloc ((0, ∞) × (0, ∞)).

for a.a. τ > 0.

2

Proposition 4.10 now allows for the following refinement of representation (4.51). Sa (t) = Sb (t/κ)e−αt/κ +

t/κ 0

Sb (τ )v(t; τ )dτ,

t > 0,

(4.54)

where the function v(t; τ ) is defined by v(t; τ ) = e−ωτ (κw0t (t − κτ ; τ ) − w0τ (t − κτ ; τ ) + ωw0 (t − κτ ; τ )) +αe−ατ e0 (t − κτ ),

t, τ > 0;

(4.55)

observe that e−αt/κ resp. αe−ατ have to be interpreted as zero in case κ = 0 or α = ∞ resp. α = ∞. The first term on the right hand side of (4.54) is just a damped dilatation of Sb (t), and therefore has the same regularity properties as Sb (t); however, the second term behaves better, we show that it is at least B(X)-continuous. Theorem 4.4 Let A be a closed linear densely defined operator in X and b, c ∈ L1loc (⺢+ ) Laplace transformable. Assume 2,1 (i) c(t) is a completely positive function, with creep function k ∈ Wloc (0, ∞);

4. Subordination

113

(ii) there is a resolvent Sb (t) of (4.23) of type (M, ωb ). Then the resolvent Sa (t) for (4.26) is of the form Sa (t) = Sb (t/κ)e−αt/κ + S0 (t),

t > 0,

(4.56)

where S0 (t) is B(X)-continuous on (0, ∞), and even on ⺢+ if κ > 0 and α < ∞; here κ = k0 , α = k∞ + k1 (0+). Proof: (a) We first prove the theorem for the special case b(t) ≡ 1, i.e. when A is the generator of a C0 -semigroup T (t) in X. Define {R(t)}t>0 ⊂ B(X) by means of R(t) = −

∞

τ T (τ )dτ w(t; τ ),

0

Since for x ∈ D(A) we have ∞

R(t)x = (τ T (τ )x) w(t; τ )dτ = 0

t > 0.

(4.57)

(T (τ )x + τ T (τ )Ax)w(t; τ )dτ,

(4.58)

∞ 0

and since by Proposition 4.9 ∞ |R(t)| ≤ − τ |T (τ )|dτ w(t; τ ) ≤ −M 0

= M

0

∞

τ eωb τ dτ w(t; τ )

∂ ∂ dτ w(t; μ)|μ=−ωb = −M s(t; μ)|μ=−ωb ∂μ ∂μ

= M s(t; −ωb ) ∗ r(t; −ωb ) ≤ Mε e(ωa +ε)t ,

t > 0,

holds for some constant Mε ≥ 1, it is clear from the uniform boundedness principle and Proposition 4.10 that R(t) is strongly continuous on ⺢+ . By (4.55) and (4.57) we obtain ∞ S0 (t) = T (τ )v(t; τ )dτ, t > 0, 0

hence by the measurability of v(·; ·)

t0 0

|S0 (t + h) − S0 (t)|dt

≤

∞ 0

≤ M

|T (τ )|

∞ 0

t0 0

|v(t + h; τ ) − v(t; τ )|dtdτ

t0

eωb τ 0

|v(t + h; τ ) − v(t; τ )|dtdτ → 0,

as h → 0. This implies in particular S0 ∗ Sa , Sa ∗ S0 , S0 ∗ S0 ∈ C(⺢+ ; B(X)), as direct estimates show. The relation ∞ d a (λ) = − tS T (τ ) (e−τ /ˆc(λ) /(λˆ c(λ))dτ dλ 0 ˆ ˆ ˆ = S(λ)/λ − (1/ˆ c) (λ)ˆ c(λ)S(λ) + (1/ˆ c) (λ)R(λ),

114


as well as (1/ˆ c) (λ) = (λκ + ω + λkˆ1 (λ)) = κ − tk˙ 1 (λ) then yield tSa (t) = [1 − κc(t) + (tk˙ 1 (t)) ∗ c(t)] ∗ S(t) − (tk˙ 1 ) ∗ R(t) + κR(t).

(4.59)

Similarly, the identity ˆ κR(λ) = κ(1/(λˆ c(λ)))((z − A)−2 ) ◦ (1/ˆ c(λ)) = κ(ˆ c(λ)/λ)(I − cˆ(λ)A)−2 = κλˆ c(λ)Sâ (λ)2 = κ(κ + kˆ1 (λ) + ω/λ)−1 Sâ (λ)2 yields in case κ > 0 κR(t) = (Sa ∗ Sa )(t) + (q ∗ Sa ∗ Sa )(t),

t > 0,

(4.60)

where q(t) is locally integrable on ⺢+ . The semigroup property of Sb (t) = T (t) together with (4.56) implies Sa ∗ Sa = Sa ∗ S0 + S0 ∗ Sa − S0 ∗ S0 + tT (t/κ)e−αt/κ , hence combining (4.59), (4.60) with this equation, B(X)-continuity of tS0 (t) on ⺢+ follows. In case κ > 0, α < ∞, the continuity of S0 (t) at t = 0 is a consequence of ∞ 0

v(t; τ )dτ = 1 − e−αt/κ → 0

as t → 0.

(b) Consider X = L1 (⺢+ ), A = −d/dτ with D(A) = W01,1 (⺢+ ); then A + ωb I generates a C0 -semigroup T (τ ) of type (1, ωb ) in X, which is given by (T (τ )f )(x) = eωb τ f (x − τ )e0 (x − τ ),

τ, x > 0.

Therefore the resolvent S(t) of (4.31), represented by (4.54) or (4.56) with Sb (τ ) = T (τ ), results in ∞ x (S(t)f )(x) = − (T (τ )f )(x)dτ w(t; τ ) = − eωb τ f (x − τ )dτ w(t; τ ) 0 0 x (ωb −α)t/κ = e f (x − t/κ) + eωb τ f (x − τ )v(t; τ )dτ, 0

i.e. (S0 (t)f )(x) =

x 0

eωb τ f (x − τ )v(t; τ )dτ,

t, x > 0.

B(X)-continuity of S0 (t) in this special case amounts to ∞ eωb τ |v(t; τ ) − v(t¯; τ )|dτ → 0 as t¯ → t > 0, 0

and also for t = 0 in case κ > 0. (c) We consider now the general case. Then ∞ S0 (t) = Sb (τ )v(t; τ )dτ, 0

t > 0,

(4.61)

4. Subordination

115

hence by step (b) |S0 (t) − S0 (t¯)| ≤ M

∞ 0

eωb τ |v(t; τ ) − v(t¯; τ )|dτ → 0

as t¯ → t > 0, and also for t = 0 in case κ > 0. The proof of Theorem 4.4. is complete. 2

Uniform continuity of the resolvent Sa (t) of (4.26) has a number of important consequences. For example, the variation of parameters formula (1.8) then yields mild solutions of (4.26) if only f ∈ BV (J; X) rather than f ∈ W 1,1 (J; X); Proposition 1.2 holds with ‘W 1,1 ’ replaced by ‘BV ’ in this case. Another consequence is a simple characterization of compactness of subordinated resolvents. Corollary 4.7 Let the assumptions of Theorem 4.4 be satisfied. Then the subordinated resolvent Sa (t) is compact for all t > 0 if and only if (z0 − A)−1 is compact for some z0 ∈ ρ(A) and either Sb (t) is compact for all t > 0 or 1/κ + α = ∞. Proof: If Sa (t) is compact for each t > 0 then the resolvent equation (1.6) for t → 0+ with x = (z0 − A)−1 y, z0 ∈ ρ(A) implies that (z0 − A)−1 is compact. This in turn implies (a ∗ Sa )(t) compact for all t ≥ 0, and so Sâ (λ) = 1/ˆ a(λ)(a ∗ Sa )∧ (λ) also has this property for all Re λ > ωa , since a ∗ Sa is B(X)-continuous. If 1/κ + α < ∞ we have by (4.56) Sˆ0 (λ) = Sâ (λ) − κSˆb (λκ + α) = Sâ (λ) −

κ (I − ˆb(λκ + α)A)−1 ; λκ + α

thus Sˆ0 (λ) is compact for all Re λ > ωa . The complex inversion formula for the Laplace transform eγt N →∞ 2π

S0 (t) = lim

N −N

(1 −

|ρ| iρt ˆ )e S0 (γ + iρ)dρ, N

t > 0, γ > ωa ,

then implies compactness of S0 (t) for all t > 0, since this limit exists in the uniform operator topology, thanks to uniform continuity of S0 (t). Therefore in case 1/κ + α < ∞, Sb (t) must be compact for all t > 0. Conversely, if (z0 − A)−1 is compact for some z0 ∈ ρ(A) then Sˆ0 (λ) and Sâ (λ) are compact for all Re λ > ωa , hence as above S0 (t) has this property for all t > 0. (4.56) then shows that Sa (t) is compact for all t > 0 if either 1/κ + α = ∞ or Sb (t) is compact for all t > 0. 2

4.7 Comments a) For a proof of Bernstein’s Theorem see e.g. Widder [339], which contains also Proposition 4.1. The notion of Bernstein functions has been introduced by Berg and Forst [21], although Proposition 4.2 and a representation similar to Proposition 4.3 is already in Feller [115]. Proposition 4.3 is taken from Clément and Pr¨ uss [54].

116


b) The notion of completely positive functions has been introduced by Clément and Nohel [50], [51] where also the equivalence (i) ⇔ (v) in Proposition 4.5 appears for the first time. The importance of this class of kernels has also been recognized in other fields; Berg and Forst [21] study potential kernels, Kingman [198] p-standard functions, and it turns out that up to certain normalizations these classes coincide. The equivalences of (i) ∼ (iv) appear also in Berg and Forst [21]. Proposition 4.4 is originally due to Gripenberg [152]; for the present approach see Clément and Pr¨ uss [54]. Lemma 4.1 is independently due to Friedman [120] and to Miller [241], while its last part was obtained by Reuter [290] before; see also Hirsch [182] and Gripenberg [153]. c) The subordination principle for completely positive measures was first observed by Bochner [26], its application to vector-valued Volterra equations appears in Pr¨ uss [276] where also Theorem 4.1 in a weaker form was obtained. Theorem 4.2 is due to Clément and Nohel [51] while Lemma 4.2 and Theorem 4.3 (ii) are proved in Pr¨ uss [274]. The present approach to Theorems 4.2, 4.3 and Corollary 4.4 based on the subordination principle is taken from Pr¨ uss [276]. d) In Friedman [121] it was observed that in case X is a Hilbert space, A negative definite, and a = c is nonnegative, nonincreasing, and log-convex, the resolvent S(t) of (1.1) has the property that (S(t)x, x) is nonnegative, nonincreasing, for each x ∈ X. This is implied by the representation (1.24) of S(t) via the functional calculus of selfadjoint operators in Theorem 1.1. Obviously this assertion remains valid if A is merely negative semidefinite and a(t) a completely positive function, by Proposition 4.5. e) The propagation function w(t; τ ) associated with a completely positive measure dc is the main subject in potential theory, where the semigroups in C0 (⺢) and Lp (⺢) defined by w(t; τ ) by means of ∞ f (t − s)ds w(s; τ ), t ∈ ⺢ (4.62) (T (τ )f )(t) = 0

are studied. It has been shown that a positive translation invariant C0 -semigroup in Lp (⺢) of contractions leaving invariant the subspace Lp (⺢+ ) of Lp (⺢) is given by (4.62), where w(t; τ ) is the propagation function associated with a completely positive measure; see Berg and Forst [21] and Clément and Pr¨ uss [54]. The connection of the propagation function with Volterra equations was first observed and exploited in Pr¨ uss [274] which also contains Proposition 4.9 and Theorem 4.4 in weaker form. Proposition 4.8 is taken from Clément and Pr¨ uss [54]. A physical interpretation of w(t; τ ) will be given in Section 5.4. f) The following example shows that for each k ∈ ⺞ there are k-monotone functions c(t) which are not completely positive. Example 4.1: Define c(t) = (1 − t)l for t ≤ 1, c(t) = 0 for t > 1; obviously c(t) is (l + 1)-monotone. Suppose that c(t) is also completely positive. Then there are

4. Subordination

117

k0 , k∞ ≥ 0, k1 (t) nonnegative and nonincreasing, such that k0 c + (k1 + k∞ ) ∗ c = 1. This implies k0 c(t) ˙ + (k1 + k∞ ) ∗ c(t) ˙ = −(k1 (t) + k∞ )c(0),

t ≥ 0,

and k0 c¨(t) + (k1 + k∞ ) ∗ c¨(t) = −k˙ 1 (t)c(0) − (k1 (t) + k∞ )c(0), ˙

t ≥ 0.

With c(0) = 1, c(0) ˙ = −l, c¨(0) = l(l − 1) these equations yield −k0 l = −(k1 (0+) + k∞ )

and

k0 l(l − 1) = −k˙ 1 (0+) + l(k1 (0+) + k∞ ),

hence 0 ≤ −k˙ 1 (0+) = k0 l(l − 1) − l(k1 (0+) + k∞ ) = k0 l(l − 1 − l) = −k0 l, i.e k0 l = 0 and so k1 (0+) + k∞ = 0, which contradicts k0 c + (k1 + k∞ ) ∗ c = 1. 2

It is not known whether any completely positive function is 1-regular. g) It follows from Proposition 4.5 that εδ0 + dc is completely positive whenever ε > 0 and dc is completely positive. However, the class of completely positive kernels is not closed under addition; in fact, εdt + dc is completely positive for each ε > 0 only if dc is completely positive and c(t) is a creep function with c1 (t) convex. On the other hand, the class of creep functions c(t) with c1 (t) log-convex - according to Proposition 4.6 dc is then completely positive - is closed under addition. It would be interesting to know whether the class of creep functions c(t) such that dc(t) is completely positive is closed under addition. This question arises naturally in viscoelasticity; see Section 5. In the same connection extensions of Lemma 4.2 and 4.3 in this direction would be of importance. h) The subordination principle for integral resolvents mentioned at the end of Section 4.3 reads as follows. Theorem 4.5 Let A be a closed linear densely defined operator in X and b, c ∈ ∞ L1loc (⺢+ ) such that 0 |b(t)|e−βt dt < ∞ for some β ∈ ⺢. Assume (i) c(t) is a completely positive function; (ii) there is an integral resolvent Rb (t) of (4.23) such that |Rb (t)| ≤ ϕb (t)eωb t for a.a. t ≥ 0, where ωb ≥ 0 and ϕb ∈ L1 (⺢+ ). Then there is an integral resolvent Ra (t) for (4.26). If moreover ωa is defined according to (4.27) then |Ra (t)| ≤ ϕa (t)eωa t

for a.a. t ≥ 0, for some ϕa ∈ L1 (⺢+ ).

(4.63)

118


For the proof we only verify (K2) of Theorem 1.6; (K1) follows from the definition of ωa . With Ka (λ) = (1/ˆ a(λ) − A)−1 we obtain

(λ − ωa )

n

|Lnλ Ka (λ)|

n≥0

≤

∞ 0

0

(λ − ωa )

n≥0

∞

n 0

|Rb (τ )|Lnλ e−τ /ˆc(λ) dτ

(λ − ωa )n Lnλ e−τ /ˆc(λ) dτ

n≥0 ∞

=

ϕb (τ )eωb τ

≤

ϕb (τ )eωb τ e−τ /ˆc(ωa ) dτ = |ϕb |1 < ∞,

λ > ωa .

Hence Theorem 1.6 implies existence of the integral resolvent Ra (t) for (4.26) with property (4.63). By means of the propagation function a representation of the integral resolvent can be given as well. Ra (t) =

d dt

∞ 0

Rb (τ )w(t; τ )dτ,

t ≥ 0;

observe that an invariance result similar to Corollary 4.6 is valid for the integral resolvent, too, as soon as 0 ∈ D, or more generally, for Ra (t)/c(t). i) A compactness result similar to Corollary 4.7 is derived in Hannsgen and Wheeler [168] in case X is a Hilbert space, A is negative definite with A−1 compact, and a(t) 2-monotone. Then S(t) is compact for each t > 0 provided −a(0+) ˙ = ∞. j) If the kernel c(t) appearing in the subordination principle is completely monotonic rather than completely positive, much more can be said about the regularity behaviour of the resolvent S(t). A typical result in this direction, using the framework of resolvents, is the following result which is taken from Pr¨ uss [277]. It deals with the second order case, i.e. b(t) ≡ t, which is of interest in viscoelasticity; see Section 5. Theorem 4.6 Suppose A generates a cosine family T (t) in X, and let a(t) be a Bernstein function; define χ(ρ) =

−Imda(iρ) , ρReda(iρ)

ρ ∈ ⺢,

and let S(t) denote the resolvent for (4.26). Then (i) if limρ→∞ ρχ(ρ) < ∞, then S(t) is an analytic resolvent for (4.26); (ii) if limρ→∞ [χ(ρ) log ρ] = 0, then S ∈ C ∞ ((0, ∞); B(X, XA )); (iii) if t0 = 2limρ→∞ χ(ρ) log ρ > 0, then S ∈ C n ((tn , ∞); B(X)) and AS ∈ C n ((tn+2 , ∞); B(X)), where tn = (n + 1)t0 .

(4.64)

4. Subordination

119

Observe that by Proposition 3.10, 1/ρ a∞ /2ρ2 + 0 ta1 (t)dt cχ(ρ) ≤ ≤ c−1 χ(ρ), 1/ρ a0 − 0 ta˙ 1 (t)dt

ρ ∈ ⺢,

(4.65)

for some constant c > 0. Therefore examples for (i) of Theorem 4.6 are a0 > 0 or a1 (t) ∼ tα−1 as t → 0, for some α ∈ (0, 1), while (ii) resp. (iii) are implied by a0 = 0, a1 (0+) < ∞, and −a˙ 1 (t) ∼ tα−1 resp. a˙ 1 (t) ∼ log t as t → 0. For an equation occuring in one-dimensional viscoelasticity, a result similar to Theorem 4.6 was first proved by Hrusa and Renardy [185]. Desch and Grimmer [84] consider hyperbolic systems of first order partial integrodifferential equations in one space dimension, and present conditions for gradual smoothing of the solutions in time, like (ii) and (iii) of Theorem 4.6, in an L2 -setting. While Theorem 4.6 covers linear isotropic viscoelasticity, by means of semigroup methods, Desch and Grimmer [83] obtained a corresponding result also for nonisotropic materials which, however, must be completely monotonic; cp. Section 5. Appendix: Some Common Bernstein Functions For the convenience of the reader we list below some common Bernstein functions ϕ as well as the quantities c, k, r, s, w related to them, according to Sections 4.2 and 4.5. If ϕ(λ) is a Bernstein function, recall the following relations: cˆ(λ) =

sˆ(λ; μ) =

1 , λϕ(λ)

ϕ(λ) ˆ k(λ) = 2 λ

ϕ(λ) 1 · , λ ϕ(λ) + μ

w(λ; ˆ τ) =

e−τ ϕ(λ) ; λ

the function r(t; μ) is obtained from s(t; μ) via r(t; μ) = (1 −s(t; μ))/μ; see Section 4.2. We use the standard notation for the complementary error function ∞ 2 2 erfc(x) = √ e−r dr, x ∈ ⺢, π x the gamma function

∞

Γ(x) =

e−t tx−1 dt,

x > 0,

0

the exponential integral

∞

E1 (x) =

e−t dt/t,

x > 0,

x

and the modified Bessel function of order 0 1 π I0 (x) = cosh(x cos θ)dθ, π 0

x ∈ ⺢.

120


In Table 4.1 c(t) and s(t; μ) are expressed in terms of elementary functions and integrals for diverse Bernstein functions ϕ(λ); here α ∈ (0, 1). ϕ(λ)

c(t)

s(t; μ)

1

1

(1 + μ)−1

λ √ λ

t 2 t/π

e−μt

√ 2 eμ t erfc(μ t)

1+λ

1 − e−t

(1 + μ)−1 [1 + μe−(1+μ)t ]

λ 1+λ

1+t

(1 + μ)−1 [μ + e−tμ/(1+μ) ] ∞ √ μπ −1 0 e−rt (r + (r − μ)2 )−1 dr/ r

λ+

√ λ

λ√ λ+ λ −λ

1−e

2

√ t/π + et erfc t − 1 1 + 2 t/π ∞ n=0 e0 (t − n)

2

log(1 + λ)

∞ t −τ ρ−1 [ 0e τ dτ ]dρ/Γ(ρ) 0

λα

tα /Γ(α + 1)

2

√

μ t (1 + μ)−1 eμ t/(1+μ) erfc( 1+μ ) ∞ 1 − μ n=0 (μ + 1)−(n+1) e0 (t − n) ∞ t 1 − 0 μe−μρ [ 0 e−τ τ ρ−1 dτ ]dρ/Γ(ρ) μr α−1 sin απ ∞ −rt e π μ2 +r 2α +2μr α cos απ dr 0

Table 4.1

Table 4.2 contains expressions for the creep functions k(t) and the propagation functions w(t; τ ) associated with several Bernstein functions ϕ(λ). ϕ(λ)

k(t)

w(t; τ )

1

t

e−τ

λ √ λ

1 2 t/π

e0 (t − τ ) √ erfc(τ /2 t)

1+λ

1+t

λ 1+λ

1 − e−t 1 + 2 t/π √ 2 t/π + et erfc t − 1

e−τ e0 (t − τ ) √ t √ e−τ [I0 (2 tτ )e−t + 0 e−s I0 (2 sτ )ds] √ erfc(τ /2 t − τ )e0 (t − τ ) √ ∞ r 1 − 0 e−rt−rτ /(1+r) sin(τ 1+r )dr/πr ∞ e−τ n=0 τ n e0 (t − n)/n! t −s sτ −1 e Γ(τ ) ds 0 α ∞ 1 − π −1 0 e−rt−τ r cα sin(τ rα sα )dr/πr

√ λ+ λ λ√ λ+ λ −λ

1−e

min{t, 1}

log(1 + λ)

1 − e−t + tE1 (t)

λα

t1−α /Γ(2 − α)

Table 4.2

Here we employed the abbreviations cα = cos απ and sα = sin απ.

4. Subordination

121

In Figure 4.1 the propagation function w(t; τ ) is depicted for ϕ = and ϕ(λ) = λ + λ/(1 + λ), respectively.

1

√ √ λ, ϕ = λ + λ,

1

0.75 75

0.75 75

6

w(t,r)

6

w(t,r) 0.5

0.5

0.25 .25

0.25 .25 4

4

0 0

0 0 t

1

t

1

2

2 r

2

2

3

3

r 4

4 5

5

1 0.75 75

6

w(t,r) 0.5 0.25 .25 4 0 0 t

1 2

2 r

3 4 5

Figure 4.1 Typical Propagation Functions

These Bernstein functions serve as typical representatives for the three different types of qualitative behaviour of w(t; τ ): infinite propagation speed, finite propagation speed but no wavefront, and finite propagation speed with wave front, respectively.

5

Linear Viscoelasticity

A rich source for vector-valued Volterra equations is the continuum mechanics for materials with memory, i.e. the theory of viscoelastic materials. In this section the basic concepts of this theory are introduced and the resulting boundary value problems are formulated. The discussion of the involved material functions shows that the notion of creep functions introduced in Section 4 appears here naturally. The well-posedness of some special problems which lead to Volterra equations of scalar type will be discussed, but also the limits of this class of equations become apparent. 5.1 Balance of Momentum and Constitutive Laws Consider a 3-dimensional body which is represented by an open set Ω ⊂ ⺢3 with boundary ∂Ω of class C 1 . Points in Ω (i.e. material points) will be denoted by x, y, .... Associated with this body there is a strictly positive function ρ0 ∈ C(Ω) called the density of mass. Acting forces will deform the body, and the material point x will be displaced to its new position x + u(t, x) at time t; the vector field u(t, x) is called the displacement field, or briefly displacement. The velocity of the material point x ∈ Ω at time t is then given by v(t, x) = u(t, ˙ x), where the dot indicates partial derivative with respect to t. The linearized strain in the body due to a deformation is defined by E(t, x) =

1 (∇u(t, x) + (∇u(t, x))T ), 2

t ∈ ⺢, x ∈ Ω,

(5.1)

i.e. E(t, x) is the symmetric part of the displacement gradient ∇u. A given strain-history of the body causes stress in a way to be specified, expressing the properties of the material the body is made of. The stress tensor will be denoted by S(t, x); both, E(t, x) and S(t, x) are symmetric. Let g(t, x) be an external body force field like gravity. Then balance of momentum in the body becomes ρ0 (x)¨ u(t, x) = div S(t, x) + ρ0 (x)g(t, x), t ∈ ⺢, x ∈ Ω; (5.2) in components (5.2) reads ρ0 (x)¨ ui (t, x) =

n ∂ Sij (t, x) + ρ0 (x)gi (t, x), ∂x j j=1

i = 1, 2, 3.

(5.2) has to be supplemented by boundary conditions; these are basically either ‘prescribed displacement’ or ‘prescribed normal stress (traction)’ at the surface ◦

◦

∂Ω of the body. Let ∂Ω = Γd ∪ Γs , where Γd , Γs are closed, Γd = Γd , Γs = Γs , ◦ ◦ and such that Γd ∩ Γs = ∅; let n(x) denote the outer normal of Ω at x ∈ ∂Ω. The boundary conditions then can be stated as follows. ◦

u(t, x)

= ud (t, x),

t ∈ ⺢, x ∈Γd ,

S(t, x)n(x)

= gs (t, x),

t ∈ ⺢, x ∈Γs .

◦

J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_5, © Springer Basel 1993

(5.3) 122

5. Linear Viscoelasticity

123

Observe that the functions g(t, x), ud (t, x), gs (t, x) here are considered as known. In many practical problems, however, the force field g(t, x) and the traction gs (t, x) are actually depending on u or on its derivatives u˙ and ∇u. For example if the body is subject to an external force field h which depends on the spatial position only, then f (t, x) = h(x + u(t, x)); other examples are boundary feedback laws like gs (t, x) = −γv(t, x). Suppose the body is at rest up to time t = 0, but is then exposed to forces g(t, x) and gs (t, x) and to a sudden velocity change u(0, ˙ x) = u1 (x). Taking the inner product of (5.2) with u˙ and integrating over Ω, an integration by parts yields the energy equality, formally at least. t ˙ x)dxdτ = |u(t, ˙ x)|2 ρ0 (x)dx + S(τ, x) : E(τ, |u1 (x)|2 ρ0 (x)dx Ω 0 Ω Ω t t + g(τ, x) · u(τ, ˙ x)ρ0 (x)dxdτ + gs (τ, y) · u(τ, ˙ y)dydτ. (5.4) 0

Ω

0

Ω

Since the total kinetic energy in the body at time t cannot exceed its initial value plus the work done by the acting body and surface forces, the inequality t ˙ x)dxdτ ≥ 0 S(τ, x) : E(τ, (5.5) 0

Ω

must hold for all values of t > 0, and for any choice of initials values and forces. A material is called incompressible, if there are no changes of volume in the body Ω during a deformation, i.e. if det (I + ∇u(t, x)) = 1,

t ∈ ⺢, x ∈ Ω,

(5.6)

is satisfied; otherwise the material is called compressible. (5.6) is a nonlinear constraint for the system (5.2), (5.3); for the linear theory (5.6) can be simplified to div u(t, x) = 0, t ∈ ⺢, x ∈ Ω. (5.7) In fact, if the material is linear, then u(t, x) in (5.6) can be replaced by μu, μ > 0; taking the derivative in (5.6) w.r.t. μ at μ = 0, (5.7) follows. As a consequence of the constraints (5.6) or (5.7), for incompressible materials the stress tensor S(t, x) cannot be completely determined by u and its derivatives; it is then convenient to decompose S(t, x) as S(t, x) = −π(t, x)I + Se (t, x),

t ∈ ⺢, x ∈ Ω,

(5.8)

where the scalar function π is called pressure, and Se (t, x) extra stress, which satisfies tr Se (t, x) = 0. Thus we have the relation − π(t, x) =

1 tr S(t, x), 3

t ∈ ⺢, x ∈ Ω.

(5.9)

While for compressible materials (5.9) is just a definition, for the incompressible case π(t, x) is an unknown function, which has to be determined by (5.2), (5.3)

124


and (5.7), too. For a mathematically oriented introduction to the basic concepts of continuum mechanics we refer e.g. to the monograph Gurtin [159]. To make the system complete we have to add an equation which relates the stress S(t, x) (or the extra stress for the incompressible case) to u and its derivatives; such relations will be referred to as constitutive relations or constitutive laws or stress-strain relations. Here we concentrate on linear materials. Since the stress should only depend on the history of the strain, the general constitutive law in the compressible case is given by ∞ ˙ − τ, x), t ∈ ⺢, x ∈ Ω, dA(τ, x)E(t (5.10) S(t, x) = 0

where A : ⺢+ × Ω → B(Sym{3}) is locally of bounded variation w.r.t. t ≥ 0; Sym{N } denotes the space of N -dimensional real symmetric matrices. In components the latter means Aijkl (t, x) = Ajikl (t, x) = Aijlk (t, x),

t ∈ ⺢+ , x ∈ Ω,

(5.11)

for all i, j, k, l ∈ {1, 2, 3}. The component functions Aijkl (t, x) are called the stress relaxation moduli of the material. Alternatively, (5.10) can be written in the form ∞ dK(τ, x)S(t − τ, x), t ∈ ⺢, x ∈ Ω, (5.12) E(t, x) = 0

where K : ⺢+ × Ω → B(Sym{3}) is locally of bounded variation in t ∈ ⺢+ . The components of K are called the creep moduli of the material and also enjoy the symmetries (5.11). Relations (5.10), (5.12) yield the identity (A ∗ dK)(t, x) = (K ∗ dA)(t, x) = tI,

(5.13)

where I denotes the identity tensor. Let us consider some special cases which have been studied extensively in classical continuum mechanics. (i) Ideally elastic solids Classical elasticity postulates the stress-strain relations S(t, x) = A∞ (x)E(t, x),

t ∈ ⺢, x ∈ Ω,

with A∞ ∈ L∞ (Ω; B(Sym{3})) positive definite, uniformly in Ω. This corresponds to the stress relaxation tensor A(t, x) = tA∞ (x),

t > 0, x ∈ Ω,

and to a creep tensor of the form K(t, x) = K0 (x),

t > 0, x ∈ Ω,


125

where A∞ (x)K0 (x) = K0 (x)A∞ (x) = I,

x ∈ Ω.

(ii) Ideally viscous fluids For such fluids the stress strain relations are ˙ x), S(t, x) = A0 (x)E(t,

t ∈ ⺢, x ∈ Ω,

with A0 ∈ L∞ (Ω; B(Sym{3})) positive definite, uniformly in Ω. Here we obtain A(t, x) = A0 (x),

t > 0, x ∈ Ω,

and K(t, x) = tK∞ (x),

t > 0, x ∈ Ω,

where A0 (x)K∞ (x) = K∞ (x)A0 (x) = I,

x ∈ Ω.

(iii) Kelvin-Voigt solids Such materials are governed by constitutive laws of the form ˙ x) + A∞ (x)E(t, x), S(t, x) = A0 (x)E(t,

t ∈ ⺢, x ∈ Ω,

with A0 , A∞ ∈ L∞ (Ω; B(Sym{3})) positive definite, uniformly in Ω. This gives A(t, x) = A0 (x) + tA∞ (x),

t ∈ 0, x ∈ Ω.

(iv) Maxwell fluids This class of materials is characterized by the stress-strain relation ∞ ˙ − s, x)ds, t ∈ ⺢, x ∈ Ω, S(t, x) = μe−μs A1 (x)E(t 0

with A1 ∈ L∞ (Ω; B(Sym{3})) positive definite, uniformly in Ω, and μ > 0. In this case we have A(t, x) = (1 − e−μt )A1 (x), t > 0, x ∈ Ω. 2

A material is called homogeneous if ρ0 , A, K do not depend on the material points x ∈ Ω. It is called isotropic if the constitutive law is invariant under the group of rotations. It can be shown that the general isotropic stress relaxation tensor is given by Aijkl (t, x) =

1 (3b(t, x) − 2a(t, x))δij δkl + a(t, x)(δik δjl + δil δjk ); 3

where δij denotes Kronecker’s symbol; similarly Kijlk (t, x) =

1 1 1 1 ( l(t, x) − k(t, x))δij δkl + k(t, x)(δik δjl + δil δjk ). 3 3 2 4

126


This results in the constitutive laws ∞ da(τ, x)E˙ (t − τ, x) S(t, x) = 2 0 ∞ 1 ˙ − τ, x), + I (3db(τ, x) − 2da(τ, x))tr E(t 3 0 1 ∞ E(t, x) = dk(τ, x)S(t − τ, x) 2 0 ∞ 1 1 1 + I ( dl(τ, x) − dk(τ, x))tr S(t − τ, x). 3 3 2 0 Taking traces in (5.14) we obtain ∞ ˙ − τ, x), tr S(t, x) = 3 db(τ, x)tr E(t 0 1 ∞ tr E(t, x) = dl(τ, x)tr S(t − τ, x). 3 0

(5.14)

(5.15)

Introducing the deviatoric stress and the deviatoric strain Sd (t, x) = S(t, x) −

1 I tr S(t, x), 3

Ed (t, x) = E(t, x) −

1 I tr E(t, x), 3

(5.16)

(5.14) and (5.15) yield Sd (t, x) = 2 Ed (t, x) =

1 2

∞

0

0

da(τ, x)E˙d (t − τ, x),

∞

dk(τ, x)Sd (t − τ, x).

(5.17)

Thus the kernels b and l describe the behaviour of the material under compression, while a and k determine its response to shear. Therefore, db is called compression modulus and da shear modulus ; from (5.15) resp. (5.17) there follow the identities (da ∗ k)(t) = t = (db ∗ l)(t),

t > 0.

(5.18)

In general, a and b are independent functions, however, if b(t, x) = βa(t, x) for some constant β > 0 then the material is called synchronous. For isotropic incompressible materials the corresponding stress-strain relation becomes ∞ S(t, x) = −π(t, x)I + 2 da(τ, x)E˙ (t − τ, x), t ∈ ⺢, x ∈ Ω, (5.19) 0

hence in virtue of the incompressibility conditon (5.7) 1 π(t, x) = − tr S(t, x), 3

t ∈ ⺢, x ∈ Ω.

(5.20)


127

The inverse relation then reads 1 ∞ E(t, x) = dk(τ, x)Sd (t − τ, x), 2 0

t ∈ ⺢, x ∈ Ω;

(5.21)

in other words l(t, x) ≡ 0 for the incompressible case. Observe that incompressible materials can be considered as the limiting case of synchronous materials when β → ∞. Inequality (5.5) leads to the following restriction on the stress relaxation tensor which will be called dissipation inequality in the sequel. T t [dA(τ, x) : F (t − τ )] : F (τ )dτ dt ≥ 0, T > 0, F ∈ C(⺢+ ; Sym{3}). 0

0

(5.22) In other words, the tensor-valued measure dA is of positive type. A special case of this are completely monotonic relaxation tensors, i.e. dA is of the form ∞ 0 ∞ dA(t, x) = A (x)δ0 + A (x)dt + ( e−tξ A1 (ξ, x)dα(ξ))dt, t > 0, x ∈ Ω, 0

(5.23) with A0 , A∞ ∈ L∞ (Ω; B(Sym{3})) positive semidefinite three-dimensional tensors, A1 : ⺢+ → L∞ (Ω; B(Sym{3})) Borel-measurable, bounded, and positive semidefinite for each t ≥ 0, and α : ⺢+ → ⺢+ nondecreasing, α(0) = α(0+) = 0, ∞ −1 (1 + ξ) dα(ξ) < ∞. 0 In the homogeneous and isotropic case, dA will be of positive type iff da and db are of positive type, and dA will be completely monotonic iff a, b are Bernstein functions. Observe also that the stress relaxation tensors from Examples (i)∼(iv) above are completely monotonic. Summarizing, we obtain the equations for linear isothermal viscoelasticity. ∞ ρ0 (x)¨ u(t, x) = div( dA(τ, x)∇u(t ˙ − τ, x)) + ρ0 (x)g(t, x), (5.24) 0

for all t ∈ ⺢ and x ∈ Ω. For homogenenous and isotropic materials (5.24) can be simplified to ∞ u ¨(t, x) = da(τ )Δu(t ˙ − τ, x) 0 ∞ 1 + (db(τ ) + da(τ ))∇∇ · u(t ˙ − τ, x) + g(t, x), (5.25) 3 0 for all t ∈ ⺢ and x ∈ Ω, where we have put ρ0 (x) ≡ ρ0 = 1 for simplicity. If in addition the material is incompressible, then (5.25) becomes ∞ u ¨(t, x) = da(τ )Δu(t ˙ − τ, x) − ∇π(t, x) + g(t, x), (5.26) 0

∇ · u(t, x) = 0,

t ∈ ⺢, x ∈ Ω.

These equations have to be supplemented by the boundary conditions (5.3).

128


5.2 Material Functions Homogeneous isotropic linear materials are described by means of two material functions, the shear modulus da and the compression modulus db. If the material is in addition synchronous or incompressible, the only function needed is the shear modulus. In this subsection we briefly want to exhibit the physical meaning of these material functions and to motivate some of their properties which are used below. Let da be the shear modulus of a homogeneous isotropic linear material and dk the corresponding creep modulus, i.e. (da ∗ k)(t) = (a ∗ dk)(t) = t,

t > 0.

(5.27)

We will now consider the behaviour of the material in simple shearing motion. Imagine two parallel plates (of zero mass), let the space between them be filled with the material under consideration. Keep one of these plates fixed, while a parallel force f (t) is applied to the second plate. f(t)

ε(t) δ

x u(t,x)

Figure 5.1: Creep and relaxation experiments

If the distance δ of these plates is small enough then inertial effects can be neglected, and displacement is only a linear function of x, the variable orthogonal to the plates. Stress and strain then only depend on time and have only one nontrivial entry (up to symmetry) σ(t) resp. ε(t), and the following relations hold. σ(t) = (da ∗ ε)(t), ˙

ε(t) = (dk ∗ σ)(t).

(5.28)

Observe also that σ(t) = −f (t), and ε(t)δ is the horizontal displacement of the second plate. We describe now two experiments from which the material functions can be visualized. Creep Experiment: Apply a sudden but then constant force at t = 0, and observe the displacement of the plate. This means σ(t) = e0 (t), the Heaviside function, and ε(t) = k(t); thus the creep modulus k(t) describes, how the material ‘creeps’ to its final position, in this experiment. It is natural to expect that k(t) is nonnegative, nondecreasing and concave, in other words k(t) is a creep function in the terminology of Section 4.


129

k∞ 1

k(∞) k(t) k0

k0 0

t rd

solid

t rx

t

0

t

fluid

Figure 5.2: Creep response

Stress Relaxation Experiment: Displace the material at t = 0 and keep it there; observe the force f (t) = −σ(t). This means σ(t) = da(t). The responding force may have an initial Dirac measure - like in the case of a Newtonian fluid but otherwise it should be a nonnegative, nonincreasing function. Therefore a(t) itself is again a creep function.

a1(t)+a ∞

a1(t)

a0

a0

a∞

a(∞) 0

solid

t

0

t rx

fluid

t

Figure 5.3: Stress relaxation

There are two ways to classify materials, according to the behaviour of a(t) and k(t) at t = 0 or at t = ∞. Behaviour at t = 0: A material is called rigid if k0 > 0; k0 is the rigidity. A material is called viscous if a0 > 0; a0 is termed dynamic viscosity. Since a0 ·k0 = 0 by Proposition 4.4, these notions are exclusive but not exhaustive; there is a third class corresponding to the case a0 = k0 = 0. As shown in Section 4.2, a material is rigid with rigidity k0 > 0 iff a0 = 0 and a1 (0+) < ∞; we then have a1 (0+) + a∞ =

1 . k0

Similarly, a material is viscous with dynamic viscosity a0 > 0 iff k0 = 0 and k1 (0+) < ∞; we then have k1 (0+) + k∞ =

1 . a0

130


Nonrigid, nonviscous materials satisfy a0 = k0 = 0 which is equivalent to a1 (0+) = k1 (0+) = ∞. Behaviour at t = ∞: A material is called a solid if a∞ > 0; a∞ is the elasticity modulus. It is called a fluid if k∞ > 0; 1/k∞ is termed static viscosity. Since a∞ · k∞ = 0 these notions are also exclusive, but not exhaustive; there is a third class according to a∞ = k∞ = 0. Solids are characterized by k(t) bounded, i.e. by ∞ 1 k∞ = 0, k1 ∈ L1 (⺢+ ), and k(∞) = k0 + k1 (τ )dτ = . a∞ 0 Similarly, fluids are equally described by a(t) bounded, i.e. by ∞ 1 a∞ = 0, a1 ∈ L1 (⺢+ ), and a(∞) = a0 + a1 (τ )dτ = k ∞ 0 Materials which are neither solids nor fluids are characterized by a∞ = k∞ = 0 which is equivalent to a1 , k1 ∈ L1 (⺢+ ). For a fluid the mean relaxation time trx is defined by ∞ ∞ ∞ trx = tda(t)/ da(t) = k∞ ta1 (t)dt. 0

0

0

A Newtonian fluid is characterized by trx = 0; in general trx can be taken as an estimate for how fast stress in a fluid relaxes. Similarly, for a solid the mean retardation time trd is defined by ∞ ∞ trd = t(k(∞) − k(t))dt/ (k(∞) − k(t))dt 0 ∞ ∞ 0 1 = t2 k1 (t)dt/ tk1 (t)dt 2 0 0 ∞ ∞ ∞ = (a0 + a1 (t)dt)/a∞ + ta1 (t)dt/(a0 + a1 (t)dt). 0

0

0

A Hookean solid is characterized by trd = 0; the mean retardation time gives an estimate for the time the material needs to creep to its final position. Let us briefly mention some well known standard models. (i) Hookean Solid a(t) = μt,

k(t) =

1 , μ

t > 0.

a(t) = ν,

k(t) =

t , ν

t > 0.

(ii) Newtonian Fluid


131

(iii) Kelvin-Voigt Solid a(t) = ν + μt,

k(t) =

1 (1 − e−μt/ν ), μ

t > 0.

1 t + , μ ν

t > 0.

(iv) Maxwell Fluid a(t) = ν(1 − e−μt/ν ),

k(t) =

(v) Poynting-Thompson Solid a(t) = μ0 t + ν(1 − e−μt/ν ),

−1 −μμ0 t/ν(μ+μ0 ) k(t) = μ−1 e ]. 0 [1 − μ(μ0 + μ)

(vi) Power Type Materials a(t) =

tα , Γ(α + 1)

k(t) =

t1−α , Γ(2 − α)

t > 0,

where α ∈ (0, 1). Observe that in any of these examples a(t) and k(t) are Bernstein functions. Let the two basic materials - Hookean solid and the Newtonian fluid - be represented by the symbols spring and dashpot. A Kelvin-Voigt solid can then be viewed as a spring and a dashpot in parallel - in this case the forces (stresses) acting on either element are additive while the Maxwell fluid corresponds to a spring and a dashpot in series - here the displacements (strains) are additive. Mechanical engineers like to think of a linear viscoelastic material as a finite network consisting of the basic elements spring and dashpot.

spring

Maxwell fluid

Poynting-Thompson solid

d a sh p o t Kelvin-Voigt solid

Figure 5.4: Mechanical models

The resulting material functions a(t) and k(t) will then be Bernstein functions of the form N aj (1 − e−t/tj ), t > 0, (5.29) a(t) = a0 + a∞ t + j=1

132


where aj , tj > 0 for all j. More rigorously, physical chemists have derived (5.29) from various molecular models. In their approach, the number N of independent relaxation times tj corresponds to the degrees of freedom of a molecule. Therefore, for complicated materials like high polymers, N will be very large and it becomes impractical to work with expressions like (5.29); nevertheless a(t) and k(t) can still be expected to be Bernstein functions. Here the so-called fractional derivative models should be mentioned, which are frequently used to describe the behaviour of polymeric materials. The five parameter fractional derivative model is given by the stress-strain law σ(t) + β(

d μ d ) σ(t) = γ(ε(t) + α( )ν ε(t)), dt dt

t ∈ ⺢,

where α, β ≥ 0, γ > 0, 0 ≤ μ ≤ ν ≤ 1. This leads to the moduli da, dk via their Laplace transforms γ 1 + αλν da(λ) = · , λ 1 + βλμ

1 1 + βλμ dk(λ) = · ; γ 1 + αλν

hence a(t) and k(t) are Bernstein functions. Such materials are solid, with a∞ = γ, and they are rigid iff μ = ν, with rigidity k0 = β/αγ; for μ < ν they are neither rigid nor viscous. Finally, let us mention that the compression modulus db(t) enjoys the same properties as the shear modulus da(t), hence b(t) and l(t) can be expected to be creep functions or even Bernstein functions, as well. 5.3 Energy Balance and Thermoviscoelasticity Consider first a rigid body Ω ⊂ ⺢3 which is subject to temperature changes. Let ε(t, x) denote the density of internal energy at time t ∈ ⺢, q(t, x) the heat flux vector field, θ(t, x) the temperature, and r(t, x) the external heat supply. Balance of energy then reads as ε(t, ˙ x) + div q(t, x) = r(t, x),

t ∈ ⺢, x ∈ Ω,

(5.30)

with boundary conditions basically either prescribed temperature or prescribed heat flux through the boundary, i.e. ◦

= θb (t, x),

t ∈ ⺢, x ∈Γb ,

−q(t, x) · n(x) = qf (t, x),

t ∈ ⺢, x ∈Γf ,

θ(t, x)

◦

◦

◦

◦

◦

(5.31)

where Γb , Γf ⊂ ∂Ω are closed, Γb = Γb , Γf = Γf , Γb ∩ Γf = ∅ and Γb ∪ Γf = ∂Ω. For the constitutive laws we shall use the linear relations ∞ ε(t, x) = dm(τ, x)θ(t − τ, x) + ε∞ (x), t ∈ ⺢, x ∈ Ω, 0 ∞ q(t, x) = − dC(τ, x)∇θ(t − τ, x), t ∈ ⺢, x ∈ Ω, (5.32) 0


133

where m ∈ BVloc (⺢+ ; L∞ (Ω)) , and C ∈ BVloc (⺢+ ; L∞ (Ω; Sym{3})). In the isotropic and homogeneous case these relations simplify to ∞ ε(t, x) = dm(τ )θ(t − τ, x) + ε∞ , t ∈ ⺢, x ∈ Ω, 0 ∞ q(t, x) = − dc(τ )∇θ(t − τ, x), t ∈ ⺢, x ∈ Ω, (5.33) 0

where m, c ∈ BVloc (⺢+ ) are scalar functions. Thus (5.30), (5.31) becomes in this case ˙ x) = dc ∗ Δθ(t, x) + r(t, x), t ∈ ⺢, x ∈ Ω, dm ∗ θ(t, ◦

θ(t, x) = θb (t, x), t ∈ ⺢, x ∈Γb , (5.34) ◦ ∂θ dc ∗ (t, x) = qf (t, x), t ∈ ⺢, x ∈Γf . ∂n From the literature one can infer that m(t) is a creep function which is also bounded; thus m0 ≥ 0, m∞ = 0 and m1 ∈ L1 (⺢+ ). m0 = m(0+) is called ∞ the instantaneous heat capacity, while m(∞) = m0 + 0 m1 (τ )dτ is termed equilibrium heat capacity. One should have at least m(∞) > 0 (i.e. m(t) ≡ 0), but most authors also inquire m0 > 0. m1 (t) is called energy-temperature relaxation function. Concerning the function c(t), however, the literature is somewhat controversial. From Meixner [240], Gurtin and Pipkin [157], and Nunziato [261] one can expect that c(t) is a bounded creep function as well, in particular c∞ = 0 and c1 ∈ L1 (⺢+ ). ∞ c0 = c(0+) ≥ 0 is called instantaneous conductivity, c(∞) = c0 + 0 c1 (τ )dτ equilibrium conductivity and c1 is termed heat conduction relaxation function. At least on should have c(∞) > 0, and there result two different theories according to c0 > 0 or c0 = 0. Meixner [240] calls a material of relaxation type, if m(t) and c(t) are Bernstein functions, i.e. if m1 , c1 are completely monotonic. On the other hand, Clément and Nohel [51], Clément and Pr¨ uss [54], and Lunardi [226], write t c(t) = c0 − γ(s)ds, t > 0, (5.35) 0 ∞ with c0 > 0 γ(s)ds, γ(t) ≥ 0 nonincreasing; in this case c(t) is 2-monotone. Thus ∞ in this theory the equilibrium conductivity c(∞) = c0 − 0 γ(s)ds > 0 is smaller than the instantaneous conductivity, in contrast to Nunziato [261]. Next let us consider linear thermoviscoelasticity, i.e. the coupling of balances of momentum and energy. For this we have to replace the isothermal stress-strain law (5.10) by ∞ ∞ ˙ − τ, x), ˙ − τ, x) − S(t, x) = dA(τ, x)E(t dB(τ, x)θ(t (5.36) 0

and the first equation in (5.32) by ∞ ε(t, x) = dm(τ, x)θ(t − τ, x) + 0

0

∞ 0

˙ − τ, x) + ε∞ (x), dB(τ, x) : E(t

(5.37)

134


for all t ∈ ⺢, x ∈ Ω. Here the second term in (5.36) corresponds to the stress generated by a nonuniform temperature-history, while the second term in (5.37) represents the conversion of strain energy into internal energy. In case A(t, x) = tA∞ (x), B(t, x) = tB ∞ (x), m(t, x) = m0 (x), t > 0, then (5.36) and (5.37) yield the corresponding relations for thermoelasticity. The equations of linear anisotropic thermoviscoelasticity read now ∞ ρ0 (x)¨ u(t, x) = div[ dA(τ, x)∇u(t ˙ − τ, x) ∞ 0 ˙ − τ, x)] + ρ0 (x)g(t, x), − dB(τ, x)θ(t 0 ∞ ∞ ˙ − τ, x) = div[ dm(τ, x)θ(t dC(τ, x)∇θ(t − τ, x)] (5.38) 0 0 ∞ − dB(τ, x) : ∇¨ u(t − τ, x) + r(t, x), 0

for t ∈ ⺢ and x ∈ Ω. Observe that B ∈ BVloc (⺢+ ; L∞ (Ω; Sym{3})). (5.38) has to be supplemented with the boundary conditions (5.3) and (5.31). In the case of isotropic materials, the deviatoric stress Sd (t, x) is only temperature-dependent through the shear modulus da, however, this is a second order effect which has to be neglected in the linear theory. The normal stresses are on the other hand influenced by the change in volume due to change in temperature. This leads to the constitutive law ∞ ∞ ˙ − τ − s, x)}, (5.39) ˙ tr S(t, x) = 3 db(τ, x){tr E(t − τ, x) − dα(s, x)θ(t 0

0

i.e.

B(t, x) = I

∞ 0

db(τ, x)α(t − τ, x),

t ∈ ⺢+ , x ∈ Ω.

Here α(t, x) represents the expansion of a volume element induced by a sudden temperature change; α is expected to be a creep function, which in addition is bounded. We summarize the equations of linear thermoviscoelasticity for homogeneous isotropic materials; w.l.o.g. let ρ0 (x) = ρ0 = 1. ∞ ∞ 1 u ¨(t, x) = da(τ )Δu(t ˙ − τ, x) + (db(τ ) + da(τ ))∇∇ ◦ u(t ˙ − τ, x) 3 0 0 ∞ ∞ ˙ − τ − s, x)] + g(t, x) − db(τ )[ dα(s)∇θ(t (5.40) 0 0 ∞ ∞ ˙ − τ, x) = dm(τ )θ(t dc(τ )Δθ(t − τ, x) 0 0 ∞ ∞ − db(τ )[ dα(ρ)∇ · u ¨(t − τ − ρ, x)] + r(t, x), 0

0


135

for all t ∈ ⺢ and x ∈ Ω. Observe that the equations of classical linear thermoelasticity are obtained from (5.40) with a(t) ≡ a∞ t, c(t) ≡ c0 ,

b(t) ≡ b∞ t, α(t) ≡ α0

m(t) ≡ m0 ,

for all

t > 0;

(5.41)

the equations then read u ¨ = a∞ Δu + (b∞ + a∞ /3)∇∇ · u − b∞ α0 ∇θ + f m0 θ˙ = c0 Δθ − b∞ α0 ∇ · u˙ + r.

(5.42)

Here the constants a∞ , b∞ , m0 , c0 , α0 are all positive and are the shear modulus, compression modulus, heat capacity, heat conductivity, coefficient of thermal expansion, respectively. 5.4 Some One-dimensional Problems This subsection is devoted to some specific problems in isothermal viscoelasticity which lead to one-dimensional boundary values problems. (i) Simple Shear. The Rayleigh Problem Consider a homogeneous isotropic viscoelastic fluid filling a halfspace, and which is at rest up to time t = 0. But then the bounding plane is suddenly moved tangentially to itself with constant speed 1. The induced velocity field of the fluid will be parallel to that of the boundary and will depend only on t and the distance x > 0 to the bounding plane. Assuming non-slip boundary conditions and the stress-strain relation of homogeneous isotropic materials (5.14) or (5.19), we obtain the following boundary value problem for the velocity w(t; x) of the fluid: t wt (t; x) = da(τ )wxx (t − τ ; x), t, x > 0, 0

w(t; 0) = 1,

t > 0,

w(0; x)

x > 0.

= 0,

(5.43)

This is Stokes’ first problem or the Rayleigh problem of viscoelasticity. Essentially the same problem arises if the material is a solid, when the boundary is suddenly displaced; w(t; x) then means the displacement in the body. Let h(λ; x) denote the Laplace transform of w(t; x) w.r.t. t > 0; a simple computation yields h(λ; x) =

1 x exp(− ), λ a ˆ(λ)

λ, x > 0.

(5.44)

Assume that a(t) is a creep function with a1 (t) log-convex; then by Lemma 4.2, there is a completely positive function c(t) such that a ˆ(λ) = cˆ(λ), λ > 0. Thus w(t; x) must be the propagation function associated with c(t); c.p. Section 4.5.

136


We may now apply Propositions 4.9 and 4.10 for the description of the solution of the Rayleigh problem. For this purpose we have to determine the characteristic numbers κ, ω, α of c(t), defined in (4.35), in terms of those of a(t). With ϕ(λ) = 1/ a ˆ(λ) and by the relations limλ→0+ λˆ a1 (λ) = a1 (∞) = 0, limλ→∞ λˆ a1 (λ) = ˆ˙1 (λ) = a˙ 1 (0+), we obtain a1 (0+), and limλ→∞ λa ω

=

lim ϕ(λ) = 0, λ→0+

κ α

= =

λ→∞

0 (a∞ + a1 (0+))−1/2 ∞ lim (ϕ(λ) − κλ) = −a˙ 1 (0+)

if a0 > 0 if a0 = 0,

λ→∞

if κ = 0 if κ > 0.

lim ϕ(λ)/λ =

2(a∞ +a1 (0+))3/2

By Proposition 4.10 the velocity field w(t; x) admits the decomposition w(t; x) = w0 (t − κx; x) + e−αx e0 (t − κx),

t, x > 0,

1,1 where w0 is continuous and even of class Wloc , provided κ = 0 or a˙ 1 (t) is locally absolutely continuous on (0, ∞). Since w(t; x) = 0 iff t ≤ κx, the perturbation propagates with speed 1/κ into the material; this speed is finite iff a0 = 0 and a1 (0+) < ∞, i.e. √ iff the material is rigid, according to the classification in Section 5.2, and κ = k0 , where k0 denotes the rigidity of the material. If α < ∞ then the wave w(t; x) exhibits a wave front, i.e. a discontinuity at t = κx, which is damped exponentially with exponent α w.r.t. x > 0. There is no wave front iff α = ∞, i.e. if κ = 0 or −a˙ 1 (0+) = ∞; the wave front is undamped iff α = 0, i.e. iff −a˙ 1 (0+) = 0, κ > 0 which corresponds to the purely elastic case a(t) ≡ a∞ t. Observe the coexistence of finite wave speed and absence of the wave front in case a0 = 0, a1 (0+) < ∞ but −a˙ 1 (0+) = ∞. We now consider briefly the general problem t ut (t, x) = da(τ )uxx (t − τ, x) + f (t, x), t, x > 0 0

u(t, 0) = g(t), u(0, x)

= u0 (x),

t > 0,

(5.45)

x > 0.

Let X denote any of the spaces C0 (⺢+ ) or Lq (⺢+ ), 1 ≤ q < ∞, and define A by means of (Au)(x) = u

(x), x > 0, where D(A) = {u ∈ X : u , u

∈ X, u(0) = 0}. It is well known that A generates a bounded cosine family Co(τ ) in X which is given by (Co(τ )u)(x) =

1 (u(x + τ ) + u(|x − τ |)sgn(x − τ )), 2

τ, x > 0.

Therefore, by Theorem 4.3 we may conclude that (5.45) admits a bounded resolvent S(t) in X, if a(t) is a creep function such that a1 (t) is log-convex, in particular


137

if a(t) is a Bernstein function. Via the representation (4.50) of S(t) and the variation of parameters formula, the solution of (5.45) can be represented explicitly in terms of the propagation function w(t; τ ), i.e. the solution of the Rayleigh Problem. (ii) Torsion of a Rod Consider a rod of length l and of uniform circular cross section with radius r > 0 made of a homogeneous isotropic viscoelastic solid. Let the left end (x = 0) of the rod be fixed, and let at the right end (x = l) of the rod a tip mass be attached, with moment of inertia β ≥ 0 around the axis of the rod, which is subject to a torque. After normalization this leads to the following initial boundary value problem for the angular displacement Θ(t, x) of the rod. Θt (t, x) =

t 0

da(τ )Θxx (t − τ, x) + h(t, x),

= 0, Θ(0, x) = Θ0 (x), t > 0, x ∈ (0, 1), t βΘt (t, 1) = − da(τ )Θx (t − τ, 1) + g(t). Θ(t, 0)

(5.46)

0

Here h(t, x) contains the given strain history as well as a probably also present distributed torque, and similarly g(t) corresponds to the given boundary strain history and the torque applied to the tip mass. Observe that such torsional problems only involve the shear modulus da, the compression modulus does not enter the analysis, at least not to first order. We concentrate now on the case β > 0; for β = 0 see (iii) below. Let X = L2 (0, 1) × ⺓ and consider the operator A in X defined by Au = A(Θ, ϑ) = (Θ

, −β −1 Θ (1)), D(A) = {(Θ, ϑ) ∈ X : Θ ∈ W

2,2

u ∈ D(A),

(0, 1), Θ(0) = 0, Θ(1) = ϑ}.

With respect to the inner product (u1 , u2 )X =

1 0

Θ1 (x)Θ2 (x)dx + βϑ1 ϑ2 ,

this operator is selfadjoint and negative definite, as can be shown in the usual way, hence generates a bounded cosine family. After an integration, with u(t) = (Θ(t, ·), Θ(t, 1)), (5.46) can be written in the abstract form (1.1) where f (t) = t t (Θ0 (·) + 0 h(s, ·)ds, Θ0 (1) − β −1 0 g(s)ds). Thus we may apply Theorem 4.3 to obtain a bounded resolvent, if the kernel a(t) is a creep function with a1 (t) logconvex. Alternatively, we may also apply Corollary 1.2 since we are in a Hilbert space setting, to the result that (5.46) admits a bounded resolvent if the kernel a(t) is merely a creep function with da(t) of positive type. (iii) Simple Tension. The Tensile Modulus As in (ii) we consider a viscoelastic rod which is fixed at the left end (x = 0), but

138


now is subject to a force in axial direction at the right end (x = l). Then the stress tensor to first order has only one nontrivial component σ(t, x) = S11 (t, x); this situation is referred to as a problem of simple tension. Let da, db denote the shear and compression moduli of the material and let dk and dl be defined according to (5.18). The stress-strain relations (5.14) then imply E=

1 1 1 1 dk ∗ S + I ( dl − dk) ∗ σ, 2 3 3 2

hence Eij ≡ 0 for i = j, E11 = 13 dk ∗ σ + 19 dl ∗ σ, and E22 = E33 = 13 ( 13 dl − 12 dk) ∗ σ, where E = (Eij ). With ε = E11 this yields the stress-strain relations ε=

1 1 (dk + dl) ∗ σ, 3 3

σ = de ∗ ε, ˙

(5.47)

where the tensile modulus de(t) is defined by 3 1 db(λ) da(λ) de(λ) = =9 , λ dk(λ) + dl(λ)/3 3db(λ) + da(λ)

λ > 0.

(5.48)

Observe that de is at least a completely positive measure if k, l are creep functions or, equivalently da, db are completely positive. However, it seems not to be clear that e(t) is also a creep function without further assumptions on a, b, although physically it should be. On the other hand, by Lemma 4.3 we see that e(t) is even a Bernstein function if a, b ∈ BF or k, l ∈ BF. If the material is synchronous, then b(t) = βa(t) for some β > 0, hence e(t) = [9β/(3β + 1)]a(t). The equations of motion for the axial extension of a viscoelastic rod after normalization become t ut (t, x) = de(τ )uxx (t − τ, x) + h(t, x), 0

u(t, 0) = 0,

t 0

de(τ )ux (t − τ, 1) = g(t),

u(0, x) = u0 (x),

(5.49)

t > 0, x ∈ (0, 1).

The linear operator defined by Au(x) D(A)

= u

(x),

x ∈ [0, 1], u ∈ D(A),

= {u ∈ X : u , u

∈ X, u(0) = 0, u (1) = 0}

is well known to generate a bounded cosine family in any of the spaces Lq (0, 1), 1 ≤ q < ∞ and in the subspace X0 = {u ∈ C[0, 1] : u(0) = 0} of C[0, 1]. Thus if e(t) is a Bernstein function (or merely a creep function with e1 (t) log-convex) then Theorem 4.3 shows that (5.49) admits a resolvent S(t), uniformly bounded by 1, in any of these spaces. For q = 2 we may apply Corollary 1.2 instead, to the result that a resolvent exists if de(t) merely is of positive type, and |S(t)| ≤ 1 then holds again.


139

5.5 Heat Conduction in Materials with Memory Consider the problem of heat conduction in materials with memory described in Section 5.3, where the rigid body Ω is assumed to be homogeneous and isotropic. Here we want to discuss the well-posedness of the resulting boundary value problem (5.34) in the usual function spaces. Let Xq = Lq (Ω), 1 ≤ q < ∞, X0 = CΓb (Ω) = {u ∈ C(Ω) : u |Γb ≡ 0}, and define (Aq v)(x) = Δv(x), x ∈ Ω, v ∈ D(Aq ), where Δ denotes the Laplace operator, and for 1 < q < ∞ D(Aq ) = {v ∈ W 2,q (Ω) : v |Γb ≡ 0,

∂v |Γ ≡ 0}, ∂n f

D(A0 ) = {v ∈ ∩q>1 D(Aq ) : A0 v ∈ CΓb (Ω)}, and in X1 = L1 (Ω), A1 is defined e.g. by the closure of A2 . If Ω ⊂ ⺢N is bounded, ∂Ω is smooth, and Γb ∩ Γf = ∅, then it is well known that Aq is closed linear densely defined, and σ(Aq ) ⊂ (−∞, 0] for each q ∈ {0} ∪ [1, ∞). Moreover, for each q and φ ∈ [0, π/2) there is a constant M (φ, q) such that |(z − Aq )−1 | ≤

M (φ, q) , |z|

z ∈ Σ(0, φ + π/2),

holds, i.e. Aq generates a C0 -semigroup which is analytic in ⺓+ and bounded on each sector Σ(0, φ), φ < π/2. In the Hilbert space case q = 2, the operator A2 is even selfadjoint and negative semidefinite, hence A2 generates a bounded cosine family in X2 = L2 (Ω). As motivated by thermodynamics, we assume that m(t) is a bounded creep function with m(0+) = m0 > 0, and denote by b(t) the corresponding completely positive function, i.e. the solution of dm ∗ b = 1; cp. Proposition 4.4. Without specifying more properties of c ∈ BVloc (⺢+ ) the initial value problem in Xq for (5.34) with homogeneous boundary conditions is then equivalent to u(t) = (a ∗ Aq u)(t) + f (t),

t > 0,

(5.50)

where a = b ∗ dc and f contains b ∗ r as well as the temperature history. Following the discussion of the physical properties of c(t) in Section 5.3, we divide into two cases. Case 1: c0 = c(0+) > 0; c ∈ BVloc (⺢+ ) arbitrary otherwise. This means that the instantaneous heat conductivity is nonvanishing; w.l.o.g. we may assume c0 = 1. In this case a = b + dk ∗ b, where k ∈ BVloc (⺢+ ) is continuous at t = 0; this shows that the perturbed equation (3.18) arises naturally in this context. Since b(t) is completely positive, it is of positive type. Hence the equation u(t) = (b ∗ Aq u)(t) + f (t),

t > 0,

(5.51)

140


is parabolic by Corollary 3.1. To show that b(t) is also 1-regular, write −λ

ˆb (λ) 1 (λ) m0 + (λm ˆ 1 (λ))

m0 − tdm = = , ˆb(λ) m0 + m ˆ 1 (λ) m0 + m ˆ 1 (λ)

Re λ > 0.

Since m1 (t) is 1-monotone and belongs to L1 (⺢+ ) by assumption, we have 1 (λ) |≤ | tdm

∞ 0

−tdm1 (t) =

∞ 0

m1 (t)dt < ∞,

Re λ > 0.

On the other hand, m0 + m ˆ 1 (λ) = 0 for Re λ ≥ 0, and converges to m0 as |λ| → ∞, hence is bounded from below. This proves that b(t) is 1-regular. Theorem 3.1 then shows that (5.51) admits a bounded resolvent S(t), and Theorem 3.2 yields a resolvent for (5.50). Thus in case c0 = c(0+) > 0, (5.34) is well-posed in any space Xq , q ∈ {0} ∪ [1, ∞). Moreover, if b(t) is in addition 2-regular, e.g. if m1 (t) is also convex, then Theorem 3.3 shows that (5.50) has the maximal regularity property of type C α in any of the spaces Xq . Observe that classical heat conduction belongs to Case 1, since then m(t) ≡ m0 and c(t) ≡ c0 for t > 0. Case 2: m, c ∈ BF bounded; materials of relaxation type. Having the case c0 = 0 in mind, one cannot expect well-posedness in each space Xq since m(t) ≡ m0 , c(t) = (1 − eαt )/α, α > 0 corresponds to the damped wave equation utt + αut = Δu + g, which for q = 2 is not well-posed in Xq , at least for dimensions N > 1, Aq then does not generate a cosine family. Therefore, we restrict to q = 2 in this case. Well-posedness in L2 (Ω) of (5.50) is an easy consequence of Corollary 1.2. 1,1 In fact, since m0 > 0, and m1 ∈ L1 (⺢+ ) we obtain b ∈ Wloc (⺢+ ), and even ˙b ∈ L1 (⺢+ ); therefore a ∈ BV (⺢+ ) and Re da(iρ) = =

c0 + cˆ1 (iρ) Re (iρˆb(iρ)dc(iρ)) = Re m0 + m ˆ 1 (iρ) [c0 + Re cˆ1 (iρ)][m0 + Re m ˆ 1 (iρ)] + Im cˆ1 (iρ) Im m ˆ 1 (iρ) ≥ 0, 2 | m0 + m ˆ 1 (iρ) |

and so Corollary 1.2 yields a resolvent S(t) for (5.50) in L2 (Ω), bounded by 1, if only m, c ∈ CF are bounded, m0 > 0, and m1 , c1 are of positive type. However, if m, c ∈ BF we may also apply the results of Section 4, in particular Theorems 4.3 and 4.4. In fact, by Lemma 4.3 1 λ

cˆ(λ) ˆb(λ)dc(λ) = cˆ(λ)/ ˆb(λ)/λ


141

is the Laplace transform of a Bernstein function e(t); observe that 1 ∗ b ∈ BF by Proposition 4.6. Therefore, Theorem 4.3 (ii) as well as Theorem 4.4 apply, hence the resolvent S(t) exists and is of the form S(t) = Co(t/κ)e−αt/κ + S0 (t),

t > 0,

where Co(t) denotes the cosine family generated by A2 . For the characteristic numbers κ, α, ω (cp. Section 4.5) we obtain

0 if c0 > 0 ˆ κ = lim 1/λ b(λ)dc(λ) = m0 /c1 (0+) if c0 = 0 λ→∞ ω α

m0 + m ˆ 1 (λ) 1/2 −1/2 lim (ˆb(λ)dc(λ)) = lim (λ ) = 0, λ→0+ c0 + cˆ1 (λ) m0 + m ˆ 1 (λ) 1/2 m0 1/2 −1/2 = lim (ˆb(λ)dc(λ)) − κλ = lim [(λ ) −( ) λ] λ→∞ λ→∞ c0 + cˆ1 (λ) c1 (0+)

=

λ→0+

1/2

1 m1 (0+) m c˙1 (0+) [ − 0 ] 1/2 2 (m0 c1 (0+)) c1 (0+)3/2 = ∞ otherwise. =

α

in case κ > 0,

Thus (5.34) has the finite propagation property iff c0 = 0, m0 > 0, c1 (0+) < ∞, and no wavefront iff κ = 0 or m1 (0+) − c˙1 (0+) = ∞. One of the motivations of the paper of Gurtin and Pipkin [157] was to remove the defect of the infinite propagation speed of classical heat flow. They already observed that this corresponds to m0 > 0, c0 = 0, c1 (0+) < ∞. On the other hand, the solutions of (5.34) should have the property of positivity, i.e. if r(t, x) ≥ 0, θb (t, x) ≥ 0, qf (t, x) ≥ 0 for all t, x, then the solution θ(t, x) of (5.34) should satisfy θ(t, x) ≥ 0 for all t, x. Assume that (5.34) enjoys the property of positivity for all domains Ω ⊂ ⺢3 which are bounded with ∂Ω smooth. Since the eigenvector corresponding to the first eigenvalue λ1 = λ1 (Ω) > 0 of −Δ with say Dirichlet boundary conditions is positive, and as Ω varies, the range of λ1 (Ω) is all of (0, ∞) this implies that −1 (λdm(λ) + μdc(λ)) is completely monotonic w.r.t. λ > 0, for each μ > 0. As −1 a result we then obtain with μ → 0+ that (λdm(λ)) is completely monotonic on (0, ∞) - which is consistent with m ∈ CF - but also that 1/dc(λ) belongs to CM, as μ → ∞. This obviously implies c0 > 0, i.e. we are in Case 1; if one even assumes c ∈ CF like is the case for materials of relaxation type, then c(t) ≡ c0 > 0 is constant. By Theorem 4.2 and Corollary 4.6 we then obtain the positivity alone from m ∈ CF, but on the other hand the wave speed is in this case necessarily infinite, the problem is parabolic. Thus there is no coexistence of positivity and finite wave speed for the linear heat conduction problem. 5.6 Synchronous and Incompressible Materials In this subsection we want to discuss the well-posedness of some problems of three dimensional viscoelasticity and thermoviscoelasticity.

142


(i) Isotropic Incompressible Fluids Consider a homogeneous isotropic incompressible viscoelastic fluid which occupies a region Ω ⊂ ⺢3 . According to Section 5.1, the velocity field v(t, x) of the fluid is governed by (5.26), hence the corresponding initial value problem becomes vt (t, x) = ∇ · v(t, x) =

0

t

da(τ )Δv(t − τ, x) − ∇π(t, x) + g(t, x),

0,

t > 0, x ∈ Ω,

v(t, x) =

0,

t > 0, x ∈ ∂Ω,

v(0, x) =

v0 (x),

t > 0, x ∈ Ω, (5.52)

x ∈ Ω,

where we employed the nonslip boundary conditions; here π(t, x) denotes the hydrostatic pressure in the fluid. The kernel da(t) is the shear modulus introduced in Section 5.1; according to Section 5.2, a(t) should be at least a creep function with da(t) completely positive. The case a(t) ≡ a0 for t > 0 corresponds to a Newtonian fluid with viscosity a0 > 0, and (5.52) then becomes the well known linear Navier-Stokes system. We want to discuss the well-posedness of (5.52) in Lq (Ω)3 where 1 < q < ∞, in particular for q = 2. Define E q (Ω) ∞ C0,σ (Ω)

1,q = {∇π : π ∈ Wloc (Ω), ∇π ∈ Lq (Ω)3 }, ∞ (Ω), = {u ∈ C0∞ (Ω)3 : ∇ ◦ u = 0}, and Lqσ (Ω) = C0,σ

where the closure is taken in Lq (Ω)3 . It is known that Lq (Ω)3 = E q (Ω) ⊕ Lqσ (Ω)

(5.53)

holds topologically if Ω is open and connected, and ∂Ω is compact and of class C 1 . This is the so-called Helmholtz decomposition of Lq (Ω)3 . Let Pq denote the projection of Lq (Ω)3 onto Lqσ (Ω) along E q (Ω) associated with the Helmholtz decomposition; Pq is bounded and even orthogonal for q = 2. Let the Stokes operator Aq be defined in Xq = Lqσ (Ω) according to (Aq u)(x) = (Pq Δu)(x), with domain

x ∈ Ω, u ∈ D(Aq ),

(5.54)

D(Aq ) = {u ∈ W 2,q (Ω)3 ∩ Xq : u |∂Ω = 0}.

It is known that Aq is a closed linear densely defined operator in Xq with σ(Aq ) ⊂ ⺢− , and for each 0 ≤ φ < π/2 there is a constant M (φ, q) ≥ 1 such that |(z − Aq )−1 |Xq ≤

M (φ, q) , |z|

z ∈ Σ(0, φ + π/2).

(5.55)

Thus Aq generates a C0 -semigroup which is analytic on ⺓+ and uniformly bounded on each sector Σ(0, φ), φ < π/2. In the Hilbert space case q = 2, A2 is even


143

selfadjoint and negative semidefinite. Since Pq ∇π(t; ·) = 0 by the definition of Pq , application of Pq to (5.52) shows that this equation can be written in abstract form as t

v(t) = 0

a(t − τ )Aq v(τ )dτ + f (t),

where

f (t) = v0 +

t > 0,

(5.56)

t 0

Pq g(τ )dτ,

t > 0;

here we assumed v0 ∈ Xq . We are now in position to apply the existence results of Sections 1∼4. Since completely positive measures are of positive type by Corollary 4.1, Corollary 1.2 yields a resolvent S2 (t) in X2 which is bounded by 1. If in addition log a1 (t) is convex, we may apply Theorem 4.3 instead, as well as Theorem 4.4 to obtain the decomposition S2 (t) = Co(t/κ)e−αt/κ + S0 (t), (5.57) where Co(t) denotes the cosine family generated by A2 , and α, κ are the characteristic numbers of a(t) defined in Section 5.4(i). For q = 2, (5.56) is not well-posed in general, restrictions on the kernel a(t) are needed. Suppose a1 (t) is 3-monotone, t t−1 0 τ a1 (τ )dτ lim < ∞, (5.58) t→0 a − t τ a ˙ 1 (τ )dτ 0 0 and consider the case of a fluid a∞ = 0, a1 ∈ L1 (⺢+ ). Then we are in the situation of Corollary 3.3, to the result that the resolvent Sq (t) of (5.52) is bounded on ⺢+ , as is tS˙ q (t), and (5.56) has the maximal regularity property of type C α , by Theorem 3.3. The remark following the proof of Corollary 3.3 yields the same result also in case a∞ = 0 or a1 ∈ / L1 (⺢+ ) if the domain Ω is bounded, since then Aq is invertible. If a(t) is even a Bernstein function and (5.58) holds then it is not difficult to show that | arg a ˆ(λ)| ≤ π/2 + φ < π for λ ∈ Σ(0, π/2 + ε), ε > 0 sufficiently small, and so Theorem 2.1 shows that the resolvent Sq (t) is even analytic. This discussion covers the case when (5.52) is parabolic. (ii) Isotropic Synchronous Materials Equations (5.25) for general homogenous isotropic viscoelastic materials in three dimensions involve two kernels, the shear modulus da and the compression modulus db. A priori there are no physical reasons why these kernels should be related, and so (5.25) cannot be transformed into an equation of the form (1.1) in some function space, except for very special configurations. For example, if Ω = ⺢3 then decomposing u(t, x) as u(t, x) = ∇ϕ(t, x) + ∇ × v(t, x) where ∇·v(t, x) = 0, it is possible to reduce (5.25) to two independent equations for ϕ and v which separately only involve the kernels (4/3)da+db and da, respectively.

144


This shows the limitations of the theory of equations of scalar type. However, if the material is synchronous, i.e. if there is β > 0 such that b(t) = βa(t), t > 0, then (5.25) can be rewritten as an equation of the form (1.1). In fact, let Ω be a bounded domain with boundary ∂Ω of class C 1 and define in X = L2 (Ω)3 an operator A by means of (Au)(x) = Δu(x) + (β + 1/3)∇∇ · u(x),

x∈Ω

(5.59)

with domain D(A) = {u ∈ W 2,2 (Ω)3 : u|Γd = 0, (

∂u 2 + (∇u)T · n + n(β − )∇ · u)|Γs = 0}. ∂n 3

It is well known that the elasticity operator A is selfadjoint and negative semidefinite and 0 ∈ ρ(A) iff Γd = ∅, N (A) ⊕ R(A) = X if Γd = ∅. Thus we can write the initial value problem for (5.25) as

t

v(t) = 0

a(t − τ )Av(τ ) + f (t),

t > 0,

in the space R(A) ⊂ X and apply Corollary 1.2 or Theorems 4.3 and 4.4 to obtain well-posedness of linear isotropic viscoelasticity in L2 (Ω)3 for the case of synchronous materials, where a ∈ CF is such that a1 is log-convex, say. For X = Lq (Ω)3 , q = 2, remarks similar to the case of incompressible materials apply. (iii) Synchronous Isotropic Thermoviscoelasticity In the simplest case of thermoviscoelasticity, the equations for homogeneous isotropic materials (5.40) already involve five kernels which in general should be considered as independent. Therefore a reduction of (5.40) to a Volterra equation of scalar type is in general not accessible. However, such a reformulation is possible in the case of synchronous materials which are defined by the relations = β da, db

2

a cˆ2 = ν 2 dm ˆ,

2 = γ 2 dm , dα λ2 a ˆ

(5.60)

with some positive constants β, ν, γ. Recall that a, b, c, m, α should be creep functions with c, m, α in addition bounded. Considering the shear modulus da and the heat capacity dm as the principal quantities, (5.60) are then definitions for the remaining material functions. Observe that classical isotropic thermoelasticity is covered by (5.60); cp. with (5.41). If a(t) and m(t) are Bernstein functions and b(t), α(t), c(t) are defined by (5.60), then b, α ∈ BF. In fact, this is trivial for b(t); for α(t) we obtain ˆ ˆ α ˆ (λ) = γ m(λ) ˆ k(λ), k(λ) = 1/(λ3 a ˆ(λ)), λ > 0, hence k, α ∈ BF by Proposition 4.6 and Lemma 4.3, respectively. However, α(t) will be bounded iff dα(0) = α(∞) < ∞ which is equivalent to m(∞) < ∞ and


145

a∞ > 0; the last relation in (5.60) is consistent with physics only for solids. However, c(t) defined by (5.60) will not always be a Bernstein function again, for this to happen, a(t) and m(t) cannot be completely independent. For example, dc(λ) = dl(1/ a ˆ(λ)), dm(λ) = ν dl(1/ a ˆ(λ))/(λ a ˆ(λ)), λ > 0, with some bounded l ∈ BF is consistent with (5.60), and m, c ∈ BF are bounded if a∞ > 0, by Proposition 4.7 and Lemma 4.3, since a ˆ(λ)/λ is a complete Bernstein function, as Lemma 4.2 shows. Observe that dc(λ) ≡ 1, i.e. l(t) = e0 (t), is a possible choice and compare the discussion at the end of Section 5.5. Now suppose the material is synchronous; choose X = WΓ1,2 (Ω)3 × L2 (Ω)3 × d 2 L (Ω) with inner product 1 (z1 , z2 )X = (∇w1 : ∇w2 + (β + )(∇ · w1 )(∇ · w2 ) + v1 · v2 + ϑ1 ϑ2 )dx, 3 Ω where zi = (wi , vi , ϑi )T . Here we assume mes(Γd ) = ∅ for simplicity and use the notation WΓ1,2 (Ω)3 = {w ∈ W 1,2 (Ω)3 : w|Γd = 0}. d Define A in X by means of ⎛

⎞ v Az = ⎝ Δw + (β + 13 )∇∇ · w − βγ∇ϑ ⎠ , νΔϑ − βγ∇ · v

(5.61)

for z ∈ D(A) = {(w, v, ϑ) ∈ W 2,2 (Ω)3 × W 1,2 (Ω)3 × W 2,2 (Ω) : ∂ϑ v|Γd = w|Γd = 0, ϑ|Γb = 0, |Γ = 0, ∂n f ∂ϑ ∂w 2 + (∇w)T · n + (β − )∇ ◦ w − βγ )|Γs = 0}. ( ∂n 3 ∂n

(5.62)

Then A is precisely the classical thermoelasticity operator in X. The variables w, v, ϑ in our setting differ from the classical ones. We define w = e ∗ v,

v = u, ˙

ϑ = dl ∗ Θ,

where the kernels e(t) and l(t) are related to a(t) resp. m(t) by eˆ(λ) = a ˆ(λ), dl(λ) = dm(λ), λ > 0;

(5.63)

(5.64)

observe that e(t) is completely monotonic by Lemma 4.2 and l is a Bernstein function by Proposition 4.7, in case a, m ∈ BF. A simple computation then shows that the initial boundary value problem for (5.40) becomes t e(t − τ )Az(τ )dτ + g(t), t > 0. (5.65) z(t) = 0

146


Since A generates a C0 -semigroup of contractions, and e(t) is completely monotonic, Theorem 4.2 applies and yields a resolvent S(t) which is bounded by 1. Concerning wave propagation, observe that the characteristic numbers ω, κ, α of e(t) are important; these are precisely those which we computed in Section 5.4. 5.7 A Simple Control Problem In manufacturing polymeric materials there arises frequently the problem of glueing. For this purpose the surfaces of the pieces first have to be heated in such a way that a surface layer of prescribed depth exceeds a certain given temperature, but stays below another critical temperature, beyond which the material would be destroyed. An effective means which meets these requirements is heating by means of radiation, e.g. infrared or microwave radiation. In a very simplified model this leads to a control problem for a one-dimensional heat equation for materials with memory. It is in particular assumed that the absorption of the radiation in the material follows an exponential distribution αe−αx , where x denotes distance from the surface and α > 0 is a constant. If θ(t, x) denotes the temperature, θ∞ = 0 the (constant) temperature of the environment, and if the constitutive laws for isotropic homogeneous materials with memory (5.33) with m ∈ BF, m0 > 0, m(∞) < ∞, c(t) ≡ c0 > 0 are employed, this leads to the problem dm ∗ θ˙

= c0 θxx + i(t)αe−αx ,

t, x > 0,

θ(0, x) = θ(t, ∞) = 0, t, x > 0, c0 θx (t, 0) = βθ(t, 0), t > 0.

(5.66)

Here i(t) denotes the intensity of the radiation which serves as the control variable; it is subject to the constraints 0 ≤ i(t) ≤ i0 ,

t > 0,

(5.67)

where i0 denotes the maximal available intensity. The constant β > 0 accounts for heat radiation of the surface of the body into its enviroment. Given a temperature θ0 > 0, the surface temperature to be reached, the problem consists of two parts. Part 1: Let i(t) ≡ i0 ; find θ(t, x), and the first time 0 < t0 < ∞, such that θ(t0 , 0) = θ0 , i.e. the desired surface temperature θ0 is reached at time t = t0 . Part 2: Find a control i(t) for t ≥ t0 , subject to the constraints (5.67) such that θ(t, 0) ≡ θ0 for t ≥ t0 , i.e. the surface temperature θ0 is maintained, and determine θ(t, x). Obviously, for t0 < ∞ to exist, the maximal intensity must be large enough. But, concerning the second part, it is not at all clear whether such a control exists. However, we show that this problem is in fact uniquely solvable. To obtain an abstract reformulation, we let a ∈ CM be defined by a ˆ(λ) = −αx , x > 0. A good choice for the state space 1/λdm(λ), λ > 0, and f (x) = αe


147

is X = C0 (⺢+ ), and with Aθ = c0 θ

, D(A) = {u ∈ C02 (⺢+ ) : c0 θ (0) = βθ(0)}, (5.66) can be rewritten as θ(t) = a ∗ Aθ(t) + (a ∗ i)(t)f,

t ≥ 0.

(5.68)

With f ∗ ∈ C0 (⺢+ )∗ defined by < θ, f ∗ > = θ(0), the observation y(t) is given by y(t) = < θ(t), f ∗ >,

t ≥ 0.

(5.69)

Since A generates a bounded analytic C0 -semigroup in X and a(t) is completely monotonic, (5.68) is a well-posed parabolic problem, its solution is given by the variation of parameters formula involving the integral resolvent R(t) for (5.68) t θ(t) = R(t − τ )f i(τ )dτ, t ≥ 0. (5.70) 0

Hence the observation becomes t y(t) = c(t − τ )i(τ )dτ, 0

t ≥ 0,

(5.71)

where the scalar function c(t) is given by c(t) = < R(t)f, f ∗ >,

t ≥ 0.

(5.72)

Observe that by maximal regularity of type C η , θ will be a strong solution on ⺢+ whenever i ∈ C η (⺢+ ). It is not difficult to compute cˆ(λ); in fact, with ϕ(z)

= < (z − A)−1 f, f ∗ > α √ = √ √ √ , ( z + α c0 )( z + β/ c0 )

z ∈ Σ(0, π),

(5.73)

one obtains ˆ cˆ(λ) = < R(λ)f, f ∗ > = < (1/ˆ a(λ) − A)−1 f, f ∗ > = ϕ(1/ˆ a(λ)), Next

λ > 0.

√ √ √ 1 z = + ( c0 + β/(α c0 )) z + β, ϕ(z) α

(5.74) z ∈ Σ(0, π),

shows that 1/ϕ(z) is a complete Bernstein function, and so ϕ(z) itself is the Laplace transform of a completely monotonic function, cp. Proposition 4.6. The subordination principle Proposition 4.7 then implies that c(t) is completely monotonic as well. It is this property of the involved kernel which leads to the well-posedness of the control problem in question. t For Part 1 of the problem we see that y(t) = i0 0 c(τ )dτ , t > 0, is a Bernstein function; hence t0 < ∞ exists if and only if βθ0 < i0 ,

(5.75)

148


since y(∞) = i0 lim

t→∞

t 0

c(τ )dτ = i0 cˆ(0) = i0 ϕ(1/ˆ a(0)) = i0 ϕ(0) = i0 /β.

For Part 2 we have to solve the Volterra equation of the first kind for the function j(t) = i(t + t0 ) resulting from (5.71) with y(t + t0 ) ≡ θ0 , i.e.

t 0

c(t − τ )j(τ )dτ = i0

t0 0

(c(τ ) − c(t + τ ))dτ,

t > 0.

(5.76)

Defining the Bernstein function k(t) by dk(λ) = 1/λˆ c(λ), i.e. √ √ dk(λ) = α−1 dm(λ) + ( c0 + β/(α c0 )) m(λ) ˆ + β/λ,

λ > 0,

the unique solution of (5.76) is given by

j(t) = = = =

d t0 i0 dk(τ ) (c(s) − c(t − τ + s))ds dt 0 0 t t0 i0 dk(τ ) −c(t ˙ − τ + s))ds 0 0 t i0 dk(τ )(c(t − τ ) − c(t + t0 − τ )) 0 t i0 (1 − dk(τ )c(t + t0 − τ )). t

0

Since dk(t) ≥ 0 and c(t) ≥ 0 is nonincreasing there follows easily 0 ≤ j(t) ≤ 1, i.e. j(t) is subject to the constraints (5.67). 5.8 Comments a) For the early history of linear viscoelasticity one should consult the papers of Boltzmann [28], Maxwell [236], and Volterra [332], [333]. These papers date back to the end of the last turn of the century and contain the isothermal field theory formulation of viscoelasticity. b) For today’s theory of viscoelasticity we refer to the monographs of Bland [24], Christensen [42]; Fl¨ ugge [117], Mainardi [233], Pipkin [269], and Renardy, Hrusa, and Nohel [289], for thermoviscoelasticity see Christensen [42] and Pipkin [269], and for heat conduction in materials with memory see Gurtin and Pipkin [157], Meixner [240], and Nunziato [261]. c) The network approach to linear isothermal viscoelasticity mentioned at the end of Section 5.2 is worked out in detail in Bland [24]. For an approach based on molecular theory see e.g. Bird, Armstrong and Hassager [23].


149

d) Usually a viscoelastic material is called a solid if it is not a fluid. Our definition of a solid is slightly different from this; however, for symmetry reasons it seems to be more natural. e) The discussion of the Rayleigh problem is standard in viscoelasticity. Pipkin [269] mentions that w(·; x) is increasing if a is a Bernstein function. Renardy [288] first observed the possibility of the finite propagation speed together with absence of the wave front; see also Hrusa and Renardy [185], where the regularity of w(t; τ ) away from {t = κτ } is studied. The continuity across this line in case −a˙ 1 (0+) = ∞, a1 (t) log-convex as well as the monotonicity of w was first proved in Pr¨ uss [274]. Torsion of a rod is also one of the standard problems in viscoelasticity. It has been discussed in detail in the papers of Hannsgen, Renardy and Wheeler [166], and Hannsgen and Wheeler [167], [170], [171], in particular stabilization and destabilization by means of boundary feedbacks; see also Desch and Miller [89]. f) Composed relaxation moduli like the tensile modulus de defined in (5.48) occur in many other problems involving linear viscoelastic rods, beams and plates; see Bland [24], Hannsgen [165], and Noren [259]. Here Lemma 4.3 proves its usefulness if a(t) and b(t) are Bernstein functions, at least. It would be of interest to know whether this lemma remains true if a, b ∈ CF are such that da and db are only completely positive. g) There has been a considerable interest in heat conduction in materials with memory in the 1960s and 1970s as the many publications from this period show. The main emphasis at that time was on the derivation of the equations from basic principles and on uniqueness results as well as on wave propagation. We refer to the papers of Coleman [57], Coleman and Gurtin [58], Gurtin [158], Gurtin and Pipkin [157], Nunziato [261], [262], Meixner [240], Finn and Wheeler [116], Nachlinger and Wheeler [254], and Davis [78], [79]. The first proofs for well-posedness in the linear case for very special configurations appear in Grabm¨ uller [134]; Miller [244] contains the first general treatment based on perturbation techniques. Since then the heat equation with memory has been used as motivation and application in many research articles. h) The properties of the Laplacian in Lq (Ω), 1 < q < ∞, used in Section 5.5 are wellknown; cp. e.g. Fattorini [113] or Tanabe [319]. The analyticity of the C0 semigroups in CΓb (Ω) and in L1 (Ω) generated by the Laplacian is more recent, it is due to Stewart [317], [318], and to Amann [8], respectively. The Helmholtz decomposition of Lq (Ω)3 for bounded domains has been obtained by Fujiwara and Morimoto [126]; for N = 3 and exterior domains see Solonnikov [311], Miyakawa [253], and von Wahl [334]. The most general result which covers domains in ⺢N with compact boundary of class C 1 seems to be the paper of Simader and Sohr [304].

150


Boundedness and analyticity of the semigroup generated by the Stokes operator was proved by Giga [129] and Solonnikov [311] for bounded domains, and by Borchers and Sohr [29], and Giga and Sohr [131] for exterior domains. The selfadjointness of the elasticity operator in Section 5.6(ii) is also wellknown; see e.g. Marsden and Hughes [234] or Leis [208]. These references contain also proofs for the m-dissipativity of the thermoelasticity operator defined by (5.61) and (5.62) in Section 5.6(iii). i) Concerning the positivity of the solutions of (5.34) in Case 1 of Section 5.5 we refer to Clément and Nohel [51] and to Lunardi [226] where sufficient conditions in terms of m(t) and c(t) are given. A complete characterization of positivity seems to be unknown.

Chapter II

Nonscalar Equations

6

Hyperbolic Equations of Nonscalar Type

The basic properties of Volterra equations of nonscalar type are discussed in this section. Resolvents for such problems are introduced and their relations to wellposedness and variation of parameters formulae are studied. The latter are then used for perturbation results which yield several well-known existence theorems. The generation theorem for the nonscalar case is proved and then applied to the convergence of resolvents and to existence theorems for equations in Hilbert spaces involving operator-valued kernels of positive type. 6.1 Resolvents of Nonscalar Equations d

Throughout this section X and Y denote Banach spaces such that Y → X, and A ∈ L1loc (⺢+ ; B(Y, X)). We consider the linear Volterra equation t u(t) = f (t) + A(t − τ )u(τ )dτ, t ∈ J, (6.1) 0

where f ∈ C(J; X), J = [0, T ]. Observe that the scalar case which has been considered so far corresponds to A(t) of the form A(t) = a(t)A, where a ∈ L1loc (⺢+ ), A a closed linear densely defined operator in X, and Y = XA , the domain D(A) of A equipped with the graph norm of A. Definition 6.1 A function u ∈ C(J; X) is called (a) a strong solution of (6.1) if u ∈ L∞ (J; Y ) and (6.1) holds in X on J; (b) a mild solution of (6.1) if there are (fn ) ⊂ C(J; X) and strong solutions (un ) ⊂ C(J; X) of (6.1) with f replaced by fn , such that fn → f and un → u in C(J; X). Comparing these solution concepts with the corresponding notions for equations of scalar type (see Definition 1.1) it is apparent that ‘strong solution’ in the sense of Definition 6.1 is the natural extension of Definition 1.1 (a). However, there is no direct analog of Definition 1.1 (b) for (6.1), and so a new concept for ‘mild solution’ had to be introduced. It is clear that strong solutions are also mild ones. As before, the central object associated with (6.1) will be the resolvent S(t). However, its definition differs somewhat from the corresponding concept introduced in Section 1.1 since (S2) no longer has a meaning. This leads to a number of complications. Definition 6.2 A family {S(t)}t≥0 ⊂ B(X) is called pseudo-resolvent for (6.1) if the following conditions are satisfied. (S1) S(t) is strongly continuous in X on ⺢+ , and S(0) = I; t (S2) U (t) = 0 S(τ )dτ is leaving Y invariant, and {U (t)}t≥0 ⊂ B(Y ) is locally Lipschitz on ⺢+ ; (S3) the resolvent equations t S(t)y = y + A(t − τ )dU (τ )y, t ≥ 0, y ∈ Y, (6.2) 0


152

6. Nonscalar Hyperbolic Equations

t

S(t)y = y + 0

153

S(t − τ )A(τ )ydτ,

t ≥ 0, y ∈ Y,

(6.3)

hold. (6.2) resp. (6.3) are called the first resp. second resolvent equation for (6.1). A pseudo-resolvent S(t) is called resolvent for (6.1) if in addition (S4) for y ∈ Y, S(·)y ∈ Y a.e. and S(·)y is Bochner-measurable in Y on ⺢+ is satisfied. A resolvent or pseudo-resolvent S(t) is called a-regular, where a ∈ L1loc (⺢+ ), if a ∗ Sx ∈ C(⺢+ ; Y ) for each x ∈ X. Observe that for each y ∈ Y, the function A(·)y belongs to L1loc (⺢+ ; X); therefore the convolution S ∗Ay appearing in (6.3) is well-defined by the strong continuity of S(t) in X, even pointwise as a Bochner integral. On the other hand, the definition of the convolution A∗dU y in (6.2) needs some care, since A(t) is not assumed to be continuous. However, we may proceed as follows. For y ∈ Y the function g(t) = U (t)y is locally Lipschitz in Y by (S2), hence belongs to BVloc (⺢+ ; Y ). Since U (0) = 0 this shows that the convolution F ∗ dg for each F ∈ C(⺢+ ; B(Y, X)) is well-defined pointwise, e.g. as a Riemann-Stieltjes integral, and F ∗dg is continuous on ⺢+ , (F ∗ dg)(0) = 0. Moreover, the estimate | (F ∗ dg)(t) |≤ L(t)

0

t

| F (τ ) | dτ,

t > 0,

(6.4)

holds on ⺢+ , where L(t) denotes the smallest Lipschitz constant of g in Y on [0, t]. By means of (G0 F )(t) = (F ∗ dg)(t), t ≥ 0, the function g defines a linear operator G0 : C(⺢+ ; B(Y, X)) → C(⺢+ ; X); (6.4) shows that G0 admits a unique bounded extension G to L1loc (⺢+ ; B(Y, X)). We then define (A ∗ dg)(t) = (GA)(t),

t ≥ 0, A ∈ L1loc (⺢+ ; B(Y, X));

observe that A ∗ dg is continuous on ⺢+ , (A ∗ dg)(0) = 0, and (6.4) holds with F replaced by A. These arguments also show that (A ∗ g)(t) = (1 ∗ (A ∗ dg))(t),

t ≥ 0,

holds. The resolvent equations can therefore be rewritten as

t

U (t)y = ty + 0

A(t − τ )U (τ )ydτ, t

U (t)y = ty + 0

U (t − τ )A(τ )ydτ,

t ≥ 0, y ∈ Y ;

(6.5)

t ≥ 0, y ∈ Y.

(6.6)

Sometimes it is more convenient to work with (6.5), (6.6) instead of (6.2), (6.3). Let us show that pseudo-resolvents are always unique. Proposition 6.1 Suppose S1 (t), S2 (t) are pseudo-resolvents for (6.1). Then S1 (t) = S2 (t) for all t ≥ 0.

154

Chapter II. Nonscalar Equations

Proof: Let y ∈ Y be fixed and let Ui (t) = (6.5) and (6.6) we obtain t ∗ U2 y

t 0

Si (τ )dτ, i = 1, 2. Since Ui (t) satisfies

= U2 ∗ (U1 − A ∗ U1 )y = U2 ∗ U1 y − (U2 ∗ A) ∗ U1 y = U2 ∗ U1 y − (U2 − t) ∗ U1 y = t ∗ U1 y,

hence t ∗ U1 y = t ∗ U2 y on ⺢+ . Differentiating twice this yields S1 (t)y = S2 (t)y for all t ≥ 0, y ∈ Y, and since Y by assumption is dense in X we obtain S1 (t) = S2 (t) for all t ≥ 0. 2

For the case of scalar equations A(t) = a(t)A, a moment of reflection shows that a pseudo-resolvent is already a resolvent, and also a-regular by Proposition 1.1. Therefore Definition 6.2 is an extension of the previous case. We are mainly interested in resolvents rather than pseudo-resolvents, so let us show that under some additional restrictions on A(t) or on the space Y a pseudo-resolvent is already a resolvent. Proposition 6.2 Suppose S(t) is a pseudo-resolvent for (6.1) and assume one of the following conditions. (i) Y has the Radon-Nikodym property. (ii) There is a dense subset Z ⊂ Y such that A(t)z ∈ Y for a.a. t > 0 and A(·)z ∈ L1loc (⺢+ ; Y ) for each z ∈ Z. (iii) S(t) is a-regular, A(t) = (a ∗ dB)(t) for t > 0, where a ∈ L1loc (⺢+ ) and B ∈ BVloc (⺢+ ; B(Y, X)) is such that B(·)y has a locally bounded Radon-Nikodym derivative w.r.t. b(t) = V arB|t0 , for each y ∈ Y. Then S(t) is a resolvent for (6.1). In cases (ii) and (iii), S(t) is even strongly continuous in Y. t Proof: Let S(t) be a pseudo-resolvent for (6.1) and U (t) = 0 S(τ )dτ . Consider first case (i). Since for each y ∈ Y the function g(t) = U (t)y is locally Lipschitz in Y, and Y has the Radon-Nikodym property (c.p. Diestel [98], Chapter 6), g(t) is a.e. differentiable in Y and g˙ ∈ L∞ loc (⺢+ ; Y ). Since Y → X we obtain g(t) ˙ = S(t)y ∈ Y for all t ∈ ⺢+ \ N (y), y ∈ Y, where N (y) has Lebesgue-measure zero, and so (S4) follows. Next assume (ii), and let z ∈ Z. The second resolvent equation then shows S(t)z ∈ Y for all t ≥ 0 and S(·)z ∈ C(⺢+ ; Y ); in fact, by the remarks following Definition 6.2 the function S ∗ Az = dU ∗ Az belongs to Y and is continuous there. Thus by (S2), Uh (t) = h−1 (U (t + h) − U (t)) is locally bounded in Y and Uh (t)z → S(t)z in C(⺢+ ; Y ), hence by the Banach-Steinhaus theorem, Uh (t) → S(t) strongly in Y, uniformly on bounded intervals. This proves (S4). For the third case, fix y ∈ Y and write (S ∗ A)(t)y = ((a ∗ S) ∗ dB)(t)y =

t 0

(a ∗ S)(t − τ )h(τ )db(τ ),


155

where h(t) = dB(t)y/db(t) denotes the Radon-Nikodym derivative of the function B(t)y in X. Since a∗Sx is strongly continuous in Y, we see that (S ∗Ay)(t) belongs to Y and is continuous there. The second resolvent equation then implies (S4). 2

Note that reflexive spaces have the Radon-Nikodym property, as well as separable dual spaces. If Y is reflexive and S(t) is a pseudo-resolvent then we even have S(t)Y ⊂ Y for each t ≥ 0 and S(·)y is weakly continuous in Y ; S(t) inherits some regularity in Y from the embedding Y → X, strong continuity of S(t) in X and reflexivity of Y. Concerning (iii), observe that kernels B(t) of the form t ∞ B(t) = Bi χ(ti ,∞) (t) + B0 (s)ds 0

i=1

1 with Bi ∈ B(Y, X) and B0 (·)y ∈ Lloc (⺢+ ; X), |B0 (t)y|X ≤ b0 (t)|y|Y such that b(t) = b0 (t) + ti 0 are covered.

6.2 Well-posedness and Variation of Parameters Formulae Suppose S(t) is a pseudo-resolvent for (6.1), let f ∈ C(J; X) and u ∈ L∞ (J; Y ) be a strong solution of (6.1). Then by (6.1) and (6.3) S ∗ u = 1 ∗ u + (S ∗ A) ∗ u = 1 ∗ u + S ∗ (u − f ) = 1 ∗ u + S ∗ u − S ∗ f, hence 1 ∗ u = S ∗ f. If u is merely a mild solution, then there are fn ∈ C(J; X), fn → f in C(J; X), and strong solutions un ∈ L∞ (J; Y ) of (6.1) with f replaced by fn , such that un → u in C(J; X). The identities 1 ∗ un = S ∗ fn then imply 1 ∗ u = S ∗ f again. Thus, if u ∈ C(J; X) is a mild solution of (6.1) then S ∗ f ∈ C 1 (J; X) and d t u(t) = S(t − τ )f (τ )dτ, t ∈ J, (6.7) dt 0 i.e. the variation of parameters formula (1.8) also holds for the general equation (6.1), in particular mild solutions are again unique. Conversely, (6.7) defines a function u ∈ C(J; X) whenever f ∈ W 1,1 (J; X) and we then have t u(t) = S(t)f (0) + S(t − τ )f (τ )dt (6.8) 0 t = S(t)f (0) + dU (τ )f (t − τ ), t ∈ J. 0

156


Thus if even f ∈ W 1,1 (J; Y ) and f (0) = 0 then u ∈ C(J; Y ) is a strong solution of (6.1). Approximating f ∈ W 1,1 (J; X) with f (0) = 0 in this space by functions fn ∈ W 1,1 (J; Y ) with fn (0) = 0, it becomes apparent that (6.8) yields a mild solution of (6.1). Thus Proposition 1.2 (i), (ii), (iv) remain valid for (6.1). However, in this general framework there is no analog of (iii) of this proposition, unless the resolvent S(t) has more special properties, e.g. is a-regular in the sense of Definition 6.2. Let us summarize. Proposition 6.3 Suppose S(t) is a pseudo-resolvent for (6.1), and let f ∈ C(J; X). Then (i) If u ∈ C(J; X) is a mild solution of (6.1), then S ∗ f is continuously differentiable on J and d t u(t) = S(t − τ )f (τ )dτ, t ∈ J; (6.9) dt 0 in particular, mild solutions of (6.1) are unique. (ii) If f ∈ W 1,1 (J; X), f (0) = 0, then

t

u(t) = 0

S(t − τ )f (τ )dτ,

t ∈ J,

(6.10)

t ∈ J,

(6.11)

is a mild solution of (6.1). (iii) If f ∈ W 1,1 (J; Y ), f (0) = 0, then

t

u(t) = 0

dU (τ )f (t − τ ),

is a strong solution of (6.1). (iv) If S(t) is a-regular and f = a ∗ g, g ∈ W 1,1 (J; X), then u(t) given by (6.10) is a strong solution of (6.1). Observe that in case S(t) is a resolvent for (6.1), S(t)x is a mild solution of (6.1) with f (t) = x, and a strong solution if x ∈ Y. In this situation, the assumption f (0) = 0 in (ii) and (iii) of this proposition can be dropped. The following example shows that a resolvent S(t) for (6.1) is in general only 0-regular in which case (iv) of Proposition 6.3 is empty, and that time-regularity of f (t) alone cannot be sufficient for existence of strong solutions, in general. Example 6.1 Let a ∈ BVloc (⺢+ ) be such that da is of positive type and let s(t) = s(t; 1) denote the solution of (1.18) with μ = 1; observe that |s(t)| ≤ 1. Consider X = l2 (⺞), Y = XA , where (Ax)n = nxn , n ≥ 1, D(A) = {(xn ) ∈ X : (nxn ) ∈ l2 (⺞)}. Then both spaces X and Y are separable Hilbert spaces and d

Y → X. Define A ∈ L1loc (⺢+ ; B(Y, X)) by means of (A(t)x)n = −a(t)xn , t ≥ 0, n ∈ ⺞; (6.1) then admits a resolvent S(t) which is given by (S(t)x)n = s(t)xn , t ≥ 0, n ∈ ⺞. Suppose u(t) is a strong solution of (6.1) on an interval J = [0, T ];


157

then |f (t)|Y

≤ |u(t)|Y + |(A ∗ u)(t)|Y ≤ |u(t)|Y + |(a ∗ u)(t)|Y ≤ (1 +

T 0

|a(t)|dt)|u|Y,∞ < ∞

for all t ∈ J, i.e. f ∈ L∞ (J; Y ). Thus if f ∈ C(J; X) but f ∈ L∞ (J; Y ) there cannot be a strong solution of (6.1). 2

For a large class of equations (6.1) existence of a pseudo-resolvent S(t) already implies a-regularity of S(t). Since this class is important for applications we state this result as Proposition 6.4 Suppose A ∈ L1loc (⺢+ ; B(Y, X)) is of the form t A(t) = a(t)A + a(t − τ )dB(τ ), t > 0,

(6.12)

0

where A is a closed linear densely defined operator in X such that Y = XA and ρ(A) = ∅, and B ∈ BVloc (⺢+ ; B(Y, X)) is left-continuous and B(0) = B(0+) = 0. Then each pseudo-resolvent S(t) for (6.1) is a-regular, and it is even a resolvent if B satisfies in addition (iii) of Proposition 6.2. Proof: Let μ ∈ ρ(A) be fixed and let K(t) = −B(t)(μ − A)−1 , t ≥ 0. Then K ∈ BVloc (⺢+ ; B(X)), hence by Theorem 0.5 there is L ∈ BVloc (⺢+ ; B(X)) such that L = K − dK ∗ L = K − L ∗ dK holds; observe that K(0) = K(0+) = 0. Rewrite the first resolvent equation as Sy − y − μa ∗ Sy = (δ + dK) ∗ (A − μ)(a ∗ dU )y,

y ∈ Y,

where δ0 (t) denotes Dirac’s distribution, and convolve with δ0 − dL to the result A(a ∗ dU y) = Sy − y − dL ∗ (Sy − y − μa ∗ Sy). Since A is closed, ρ(A) = ∅ and a ∗ dU y = a ∗ Sy in X, we obtain (a ∗ Sx)(t) ∈ Y for each t ≥ 0, x ∈ X and A(a ∗ Sx)(t) is continuous in X, i.e. a ∗ S is strongly continuous in Y . 2

In the sequel, (6.1) will be said to have a main part if A(t) is of the form (6.12). In analogy to the case of scalar equations, well-posedness of (6.1) is defined as follows. Definition 6.3 (i) Equation (6.1) is called well-posed, if for each y ∈ Y there is a unique strong solution u(t; y) of t u(t) = y + A(t − τ )u(τ )dτ, t ≥ 0, (6.13) 0

158


and u(t; yn ) → 0 in X, uniformly on compact intervals, whenever (yn ) ⊂ Y and yn → 0 in X. (ii) Let a ∈ L1loc (⺢+ ). (6.1) is called a-regularly well-posed, if it is well-posed and if the equation t t u(t) = ( a(τ )dτ )x + A(t − τ )u(τ )dτ, 0

0

t ≥ 0,

(6.14)

admits a strong solution for every x ∈ X. Suppose S(t) is a resolvent for (6.1). From its definition and Proposition 6.2 it is then evident that (6.1) is well-posed, and even a-regularly well-posed if S(t) is also a-regular. Conversely, suppose (6.1) is well-posed. Then we define S(t)y = u(t; y),

t ≥ 0, y ∈ Y ;

as in Section 1.1 it then follows that S(t) extends to all of X, is strongly continuous in X, strongly measurable and locally bounded in Y, and satisfies the first resolvent equation. Thus S(t) satisfies (S1), (S4), the first part of (S3), hence also (S2); it remains to show that S(t) satisfies the second resolvent equation, or equivalently that mild solutions of (6.1) are unique. However, it does not seem possible to derive the latter from the well-posedness of (6.1) alone; further properties are needed. Proposition 6.5 (a) If (6.1) admits a resolvent S(t) then (6.1) is well-posed. (b) If (6.1) is well-posed and (ii) of Proposition 6.2 holds, then (6.1) admits a resolvent S(t). (c) Let a ∈ L1loc (⺢+ ), a ≡ 0. Then (6.1) is a-regularly well-posed if and only if there is an a-regular resolvent S(t) for (6.1). Proof: (a) is already clear. To prove (b) we are left with the verification of the second resolvent equation. Suppose (ii) of Proposition 6.2 holds, let z ∈ Z and t consider (6.1) with f (t) = 0 A(τ )zdτ. Then f ∈ C(J; X) ∩ W 1,1 (J; Y ), hence by Proposition 6.3 the function u = S ∗ f = S ∗ Az is a strong solution of (6.1). On the other hand, we have for v = Sz − z v = A ∗ Sz = 1 ∗ Az + A ∗ v, i.e. v is also a strong solution of (6.1) and so by uniqueness of strong solutions Sz − z = v = u = S ∗ Az. This shows that the second resolvent equation holds for each z ∈ Z; since by assumption Z ⊂ Y is dense, it follows also for each y ∈ Y . (c) It is clear that existence of an a-regular resolvent S(t) implies that (6.1) is a-regularly well-posed. Conversely, to prove the second resolvent equation let f (t) = a ∗ 1 ∗ Ay, y ∈ Y . Then by Proposition 6.3, u = S ∗ f = a ∗ S ∗ Ay is


159

a strong solution of (6.1); on the other hand v = a ∗ Sy − 1 ∗ ay is also a strong solution of (6.1), hence by uniqueness of strong solutions we obtain u = v, i.e. a ∗ (Sy − y − S ∗ Ay) ≡ 0,

y ∈ Y.

Titchmarsh’s theorem implies then the second resolvent equation since a ≡ 0. To show that S(t) is also a-regular, observe that by Definition 6.3 and aregularity of (6.1), a ∗ U (t) is locally Lipschitz in B(X, Y ) and (d/dt)a ∗ U y(t) = a ∗ dU (t)y is continuous in Y for each y ∈ Y. The Banach-Steinhaus theorem then implies that a ∗ dU x = a ∗ Sx is continuous in Y for each x ∈ X, i.e. S(t) is a-regular. 2

6.3 Hyperbolic Perturbation Results Consider the perturbed equation u(t) = f (t) + (A ∗ u)(t) + (B ∗ u)(t),

t ∈ J,

(6.15)

where X, Y, A(t), f (t) are as before and B ∈ L1loc (⺢+ ; B(Y, X)). Suppose (6.1) admits a resolvent S(t) and (6.15) a resolvent R(t); then by the variation of parameters formula (6.9) we have the relations R(t)y = S(t)y +

d (S ∗ B ∗ R)(t)y, dt

t ∈ ⺢, y ∈ Y,

(6.16)

R(t)y = S(t)y +

d (R ∗ B ∗ S)(t)y, dt

t ∈ ⺢, y ∈ Y.

(6.17)

Under appropriate assumptions on B(t), these identities can be used as in Section 1.4 to show that (6.16) admits an a-(pseudo-)resolvent iff (6.1) does. Theorem 6.1 Let a ∈ L1loc (⺢+ ), and suppose B ∈ L1loc (⺢+ ; B(Y, X)) is of the form B(t)y = B0 (t)y + (a ∗ B1 )(t)y, t ∈ ⺢, y ∈ Y, (6.18) where {B0 (t)}t≥0 ⊂ B(X) ∩ B(Y ) and {B1 (t)}t≥0 ⊂ B(Y, X) satisfy (i) B0 (·)y ∈ BVloc (⺢+ ; Y ) for each y ∈ Y ; B0 (·)x ∈ BVloc (⺢+ ; X) for each x ∈ X; (ii) B1 (·)y ∈ BVloc (⺢+ ; X) for each y ∈ Y . Then (6.15) admits an a-regular (pseudo-)resolvent R(t) if and only if there is an a-regular (pseudo-)resolvent S(t) for (6.1) For the proof we need the following simple Lemma 6.1 Let Xi , i = 1, 2, be Banach spaces, J = [0, a]. Then (i) Suppose {T (t)}t≥0 ⊂ B(X1 , X2 ) is strongly continuous and f ∈ BV (J; X1 ). Then u = T ∗ f is Lipschitz in X2 on J. (ii) Suppose {U (t)}t≥0 ⊂ B(X1 , X2 ) is locally Lipschitz, U (0) = 0, and f ∈ BV (J; X1 ). Then v = U ∗ df is Lipschitz in X2 on J.

160


Proof: (i) Define f (t) = 0 for t < 0, and let ϕ(t) = Var f |t−∞ ; then ϕ(t) is nondecreasing and |f (t) − f (s)| ≤ ϕ(t) − ϕ(s), Therefore

|u(t) − u(s)|

≤

∞ 0

≤ M

s ≤ t ≤ a.

|T (τ )||f (t − τ ) − f (s − τ )|dτ

∞

0

t

= M

(ϕ(t − τ ) − ϕ(s − τ ))dτ

ϕ(τ )dτ ≤ M ϕ(a)(t − s),

s ≤ t ≤ a,

s

where M = supJ |T (t)|. (ii) Define U (t) = 0 for t < 0, the convolution v = U ∗ df exists even as a Riemann-Stieltjes integral and we obtain ∞ |v(t) − v(s)| = | (U (t − τ ) − U (s − τ ))df (τ )| 0 a ≤ L(t − s) |df (τ )| = L · Var f |a0 (t − s), s ≤ t ≤ a, 0

(s) where L = sup{| U (t)−U | : 0 ≤ s ≤ t ≤ a}. t−s 2

Proof of Theorem 6.1: Suppose (6.1) admits an a-regular (pseudo-)resolvent S(t), and let U = 1 ∗ S. For each x ∈ X, S ∗ B0 x is locally Lipschitz in X, by Lemma 6.1, hence K0 = S ∗ B0 is so in B(X), as the uniform boundedness principle shows. Similarly, K0 = U ∗ dB0 is also Lipschitz on compact intervals in B(Y ), by Lemma 6.1 (ii). The same argument yields K1 = S ∗ B1 locally Lipschitz in B(Y, X) and by a-regularity of S(t), a ∗ K1 = (a ∗ S) ∗ B1 is locally Lipschitz in B(Y ). Note that neither K0 nor K1 have a jump at t = 0. Consider the equation W x = (a ∗ S)x + dK0 ∗ W x + (d(a ∗ K1 )) ∗ W x,

x ∈ X;

(6.19)

since (a ∗ S)x is strongly continuous in Y for each x ∈ X, by Corollary 0.3, there is a unique family of operators {W (t)}t≥0 ⊂ B(X, Y ) which is strongly continuous and satisfies (6.19). (W will be a ∗ R lateron.) Next we consider the equation Rx = Sx + dK0 ∗ Rx + dK1 ∗ W x,

x ∈ X;

(6.20)

since S(t) as well as dK1 ∗ W are strongly continuous in X, by Corollary 0.3 there is a unique strongly continuous solution {R(t)}t≥0 ⊂ B(X) of (6.20). Convolving (6.20) with a and comparing the result with (6.19) in the space X, we obtain W = a ∗ R by uniqueness. In the next step we consider V y = U y + dK0 ∗ V y + (d(a ∗ K1 )) ∗ V y,

y ∈Y;

(6.21)


161

since U (t) is locally Lipschitz in Y, by Corollary 0.3 there is a unique family of operators {V (t)}t≥0 ⊂ B(Y ), strongly continuous in Y but also locally Lipschitz, satisfying (6.21). Integrating (6.20) for x = y ∈ Y and from W = a ∗ R, by uniqueness of the solution of (6.21), we then obtain V y = 1 ∗ Ry for each y ∈ Y . If S(t) is even a resolvent, we may consider the equation u = Sy + dK0 ∗ u + d(a ∗ K1 ) ∗ u,

(6.22)

for a fixed y ∈ Y . Since S(·)y ∈ L∞ loc (⺢+ ; Y ), there is a unique solution u ∈ L∞ (⺢ ; Y ) of (6.22), and again by uniqueness we obtain u(t) = R(t)y a.e. + loc So far we have shown that our candidate R(t) for the resolvent of (6.15) satisfies (S1), (S2), and also (S4) if S(t) is a resolvent, and is a-regular. It remains to verify the resolvent equations; we use their integrated forms (6.5), (6.6). Here we restrict attention to the case a ≡ 0, the other case is even simpler. (6.21) implies for y ∈ Y by the definition of K0 and K1 a ∗ 1 ∗ [(A + B) ∗ V y − V y + ty] = a ∗ 1 ∗ [A ∗ U y − U y + ty] + [A ∗ a ∗ S + a ∗ 1 − a ∗ S] ∗ B ∗ V y = 0, since S(t) is a pseudo-resolvent for (6.1); hence by Titchmarsh’s theorem V y = ty + (A + B) ∗ V y,

t ≥ 0, y ∈ Y,

which is the first resolvent equation for (6.15). To prove the second resolvent equation observe that for y ∈ Y we have W ∗ (A + B)y = (a ∗ S) ∗ (A + B)y + d(S ∗ B) ∗ (W ∗ (A + B)y) by (6.19), but also (W − 1 ∗ a)y = (a ∗ S − 1 ∗ a)y + d(S ∗ B)y ∗ 1 ∗ a + d(S ∗ B) ∗ (W − 1 ∗ a)y, hence by uniqueness of the solution and in virtue of (a ∗ S − 1 ∗ a)y + d(S ∗ B)y ∗ 1 ∗ a = a ∗ (S ∗ A + S ∗ B), we may conclude W ∗ (A + B)y = W y − 1 ∗ ay,

y ∈ Y, t ∈ ⺢+ ;

since W = a ∗ R, again Titchmarsh’s theorem implies R ∗ (A + B)y = Ry − y,

y ∈ Y, t ∈ ⺢+ ,

i.e. R(t) satisfies the second resolvent equation for (6.15). Interchanging the roles of (6.1) and (6.15), the ‘only if’ part follows as well, and the proof is complete. 2

Observe that Theorem 1.2, first part, is contained in Theorem 6.1. Let us consider some special cases which combined with the results of Chapter 1 lead to first existence results for resolvents of nonscalar equations.

162


Corollary 6.1 Let A be a closed linear densely defined operator in X, set Y = XA , let a ∈ L1loc (⺢+ ), B0 , B1 as in Theorem 6.1, and let A(t) be of the form A(t) = a(t)A + (a ∗ B1 )(t) + B0 (t),

t ∈ ⺢+ .

(6.23)

Suppose (1.1) admits a resolvent. Then (6.1) admits an a-regular resolvent. For a(t) ≡ 1, t > 0, in the situation of Corollary 6.1, (6.15) is formally equivalent to u˙ = g + Au + dB0 ∗ u + B1 ∗ u, u(0) = x. (6.24) As a consequence of Corollary 6.1 we then obtain Corollary 6.2 Let A, Y, B0 , B1 be as in Corollary 6.1. Then (6.24) is well-posed if and only if A generates a C0 -semigroup in X. If this is the case, the resolvent S(t) for (6.24) is a-regular with a(t) ≡ 1, and S(·)y is continuously differentiable in X for each y ∈ Y . For a(t) ≡ t, in the situation of Corollary 6.1, (6.15) is formally equivalent to u ¨ = h + Au + dB0 ∗ u˙ + B1 ∗ u,

u(0) = x, u(0) ˙ = y.

(6.25)

For this case Corollary 6.1 yields Corollary 6.3 Let A, Y, B0 , B1 be as in Corollary 6.1. Then (6.25) is well-posed if and only if A generates a cosine family in X. If this is the case, the resolvent S(t) for (6.25) is t-regular, and S(·)y is continuously 1,1 differentiable in X for each y ∈ Y, even twice a.e. if B0 ∈ Wloc (⺢+ ; B(X)). Observe that for equations with main part one of the conditions in (i) of Theorem 6.1 on B0 (t) can be replaced by B0 = a ∗ dC, for some C ∈ BVloc (⺢+ ; B(X)); this follows from the resolvent equations. In particular, for the first order case considered in Corollary 6.2, one of the conditions in (i) is sufficient. This remark will be useful in Section 9. 6.4 The Generation Theorem In this subsection the extension of the generation theorem, Theorem 1.3, to the case of nonscalar equations is considered. This discussion is restricted to pseudoresolvents and kernels which are of exponential growth, w.o.l.g. of subexponential growth. So we let X, Y, A(t) as before and assume in addition

∞ 0

e−εt |A(t)|B(Y,X) dt < ∞

for each ε > 0,

(6.26)


163

i.e. A(t) is growing subexponentially; a ∈ L1loc (⺢+ ) is also assumed to be of subexponential growth. Suppose S(t) is a pseudo-resolvent for (6.1) and for each ε > 0 there is a constant Mε such that |S(t)|B(X) + sup (t − s)−1 |U (t) − U (s)|B(Y ) ≤ Mε eεt 0<s 0;

(6.27)

then for each y ∈ Y the resolvent equations in integrated form yield the relations ˆ ˆ (λ)y = U ˆ (λ)(I − A(λ))y ˆ (I − A(λ)) U =

y , λ2

Re λ > 0.

ˆ ˆ (λ) = S(λ)/λ ˆ With H(λ) = S(λ) and U this equation can be rewritten as ˆ ˆ λ(I − A(λ))H(λ)y = H(λ)λ(I − A(λ))y = y,

Re λ > 0.

(6.28)

Observe that {H(λ)}Re λ>0 ⊂ B(X) ∩ B(Y ) is holomorphic in both spaces, B(X) ˆ and B(Y ), and {A(λ)} Re λ>0 ⊂ B(Y, X) is holomorphic in B(Y, X). (6.28) implies ˆ ˆ that (I − A(λ)) is invertible in Y (more precisely, the part of I − A(λ) in Y is invertible), and H(λ)y =

1 −1 ˆ (I − A(λ)) y, λ

Re λ > 0, y ∈ Y.

−1 ˆ Since Y is dense in X, and H(λ) is also bounded in X, this implies that (I −A(λ)) is closable in X and

H(λ) =

1 −1 , ˆ (I − A(λ)) λ

Re λ > 0,

ˆ in X. Note that it is unclear in general, whether the operators A(λ) considered as ˆ unbounded operators in X with domain D(A(λ)) = Y are closed or even closable! However, if S(t) is in addition a-regular for some nontrivial function a, and for each ε > 0 there is a constant Nε such that |a ∗ S(t)|B(X,Y ) ≤ Nε eεt

for each t > 0,

(6.29)

holds, then a ˆ(λ)H(λ) = (a∗S)(λ) forms a holomorphic operator family in B(X, Y ). Since the zeros of a ˆ(λ) in Re λ > 0 are isolated, the latter implies that H(λ) ∈ ˆ B(X, Y ) for each Re λ > 0. (6.28) then implies that the operators A(λ) are closed in X with constant domain Y . After this preliminary discussion the extension of Theorem 1.3 to the nonscalar case can be stated as follows. Theorem 6.2 Suppose A ∈ L1loc (⺢+ ; B(Y, X)) satisfies (6.26) and a ∈ L1loc (⺢+ ) is of subexponential growth, a ≡ 0. Then (6.1) admits a pseudo-resolvent S(t) such that (6.27) holds if and only if the following conditions are satisfied. ˆ ˆ (N1) For each λ > 0 the operators I − A(λ) are injective, the part of I − A(λ) in −1 ˆ Y is surjective, and (I − A(λ)) is bounded in X and in Y ;

164


−1 /λ, λ > 0. ˆ we let H(λ) = (I − A(λ)) (N2) For each ε > 0 there is a constant Mε ≥ 1 such that

|(

d n d ) H(λ)|B(X) + |( )n H(λ)|B(Y ) ≤ Mε n!(λ − ε)−(n+1) , dλ dλ

(6.30)

for all n ∈ ⺞0 and λ > ε. If this is the case, then S(t) is a-regular and (6.29) holds if and only if ˆ ˆ (N3) for each λ > 0 the operators A(λ) in X with domain D(A(λ)) = Y are closed, and for each ε > 0 there is a constant Nε ≥ 1 such that |(

d n ) [ˆ a(λ)H(λ)]|B(X,Y ) ≤ Nε n!(λ − ε)−(n+1) , dλ

n ∈ ⺞0 , λ > ε.

(6.31)

is satisfied. Proof: (Necessity.) For the necessity it only remains to prove the estimates (6.30) and (6.31); these follow directly from Theorem 0.3 by (6.27) and (6.29). (Sufficiency.) If (N1) and (N2) hold, by Theorem 0.3 there are locally Lipschitz functions U ∈ C(⺢+ ; B(X)), V ∈ C(⺢+ ; B(Y )), U (0) = V (0) = 0 with ˆ (λ) = H(λ)/λ in X, U

Vˆ (λ) = H(λ)/λ in Y,

λ > 0,

and for each ε > 0 there is Mε such that sup (t − s)−1 |U (t) − U (s)|B(X)

0<s 0, y ∈ Y,

(6.33)

i.e. the resolvent equations hold in integrated form. Since U (t) is locally Lipschitz in X, the family Uh (t) = (U (t + h) − U (t))/h is uniformly bounded for 0 < h < 1, and bounded t > 0. For y ∈ Y we have (U ∗ A)(t)y = 1 ∗ (dU ∗ A)(t)y,

t > 0,

and (dU ∗ A)(t)y is continuous in X; see the remarks following Definition 6.2. Therefore, (6.33) shows that the functions U (·)y are continuously differentiable in X, for each y ∈ Y . Defining S(t)y = U (t)y, t ≥ 0, y ∈ Y, S(t) is bounded in X, uniformly for bounded t ≥ 0, hence extends to a strongly continuous operator family in X, i.e. (S1) is satisfied. Differentiating (6.33) yields (S3), the resolvent equations. Finally (6.32) yields (6.27).


165

Suppose in addition that (N3) holds. Then, by Theorem 0.3 there is a locally Lipschitz family {W (t)}t≥0 ⊂ B(X, Y )) with the property that for each ε > 0 there is a constant Nε such that sup (t − s)−1 |W (t) − W (s)|B(X,Y ) ≤ Nε eεt

0<s 0,

(6.34)

ˆ (λ) = a andW ˆ(λ)H(λ)/λ, λ > 0. By uniqueness of the Laplace transform, we have W (t) = (a ∗ U )(t), t > 0. Then W (t)y = (a ∗ dU )(t)y exists for each y ∈ Y and is continuous in Y . By the uniform boundedness principle W (t)x exists in Y for each x ∈ X and is strongly continuous on ⺢+ . Finally, we have W (t)x = (a ∗ dU )(t)x = (a ∗ S)(t)x for each x ∈ X, t ≥ 0, i.e. S(t) is a-regular; (6.34) implies (6.29). The proof is complete. 2

The following example shows that not every resolvent can be obtained via the Generation Theorem; even if A ∈ BV (⺢+ ; B(Y, X)) the resolvent S(t) if it exists need not be exponentially bounded. Example 6.2 Let X = l2 (⺞), A the skew-adjoint operator defined by (Ax)n = inxn , n ∈ ⺞, with D(A) = {x ∈ l2 (⺞) : (nxn ) ∈ l2 (⺞)}, and Y = XA . Define B(t) ∈ B(Y, X)) by means of n ∈ N, t ≥ 0,

(B(t)x)n = neint χ[tn ,∞) (t)xn ,

where (tn ) ⊂ ⺢+ is an increasing sequence with tn → ∞ which will be chosen later. We consider (6.1) in differentiated form, i.e. the equation u˙ = g + Au + B ∗ u,

u(0) = x.

Observe that (B(t)x)n = 0 only for finitely many n, independent of X, hence we even have B ∈ BVloc (⺢+ ; B(X, Y )); Theorem 6.1 therefore implies that a unique resolvent S(t) exists, which is even 1-regular, since A generates a bounded C0 group in X. The important point is that |B(t)|B(Y,X) ≤ 1 holds, but |B(t)|B(X) ∼ 2 et , as we shall see. The resolvent S(t) can be explicitely computed, it is given by (S(t)x)n = eint rn (t)xn , n ∈ ⺞, t > 0 where rn (t) =

∞

nj (t − jtn )2j + /(2j)!,

t ≥ 0, n ∈ ⺞;

j=0

observe that for a given t, we have rn (t) = 1 for n ≥ n(t), where n(t) is defined by n(t) = max{n : tn ≤ t}. The Laplace-transforms of rn are easily computed to the result √ rn (λ) = λ(λ2 − ne−λtn )−1 , λ > n, n ∈ ⺞; If S(t) is exponentially bounded, say by ω, then rn (λ) cannot have poles in a 2 halfplane Reλ > ω. Now choose tn > 0 such that t2n etn = n, n ∈ ⺞; then λ = tn

166


is a pole of rn (λ). Since the function xex is strictly increasing to infinity, we have (tn ) strictly increasing and tn → ∞. Thus S(t) cannot be exponentially √ bounded. It is easy to see that log n > t2n > log n − 2 log log n, i.e. tn ∼ log n, hence 2 n(t) ∼ et . This implies for the norm of B(t) in X and also in Y the asymptotic 2 relations |B(t)|B(X) ∼ |B(t)|B(Y ) ∼ et . 2

It is very unpleasant that in the sufficiency part of Theorem 6.2 one has to verify the estimates on the derivatives of H(λ) not only in X but also in Y, and maybe even for a ˆ(λ)H(λ) in the a-regular case. However, if A(t) has a main part as in (6.12) then the estimates for H(λ) are sufficient, as we show now. Corollary 6.4 Suppose A(t) is of the form t A(t) = a(t)A + a(t − τ )dB(τ ),

t > 0,

(6.35)

0

where A is a closed linear densely defined operator in X with ρ(A) = ∅, a ∈ L1loc (⺢+ ), a(t) ≡ 0, and B ∈ BVloc (⺢+ ; B(Y, X)) is left-continuous and B(0) = B(0+) = 0, where Y = XA . Assume that a and B are of subexponential growth. Then (6.1) admits a pseudo-resolvent S(t) such that for some M ≥ 1, ω > 0, |S(t)|B(X) ≤ M eωt ,

t ≥ 0,

if and only if the following two conditions are satisfied. ˆ (N1’) For each λ > ω the operators A(λ) with domain Y = D(A) are closed, ˆ I − A(λ) is bijective. −1 ˆ (N2’) H(λ) defined by H(λ) = (I − A(λ)) /λ, λ > ω, satisfies |(

d n ) H(λ)|B(X) ≤ n!M (λ − ω)−(n+1) , dλ

n ∈ ⺞0 , λ > ω.

(6.36)

If this is the case then S(t) is already a-regular. Proof: If A(t) has the form (6.35) then by Proposition 6.4 any pseudo-resolvent of (6.1) is already a-regular. Thus the necessity part as well as the last statement follow from Theorem 6.2. For the sufficiency part, it remains to show that (N2’) implies (N2), probably with λ replaced by λ + α, for some α ≥ ω. For this purpose, we fix any μ ∈ ρ(A) and let K(t) = −B(t)(μ − A)−1 , and denote by L(t) the solution of the Volterra equation L = K − dK ∗ L. Then we have (ˆ a(λ)A + a ˆ(λ)dB(λ))H(λ) = H(λ)(ˆ a(λ)A + a ˆ(λ)dB(λ)), hence

(I + dK(λ))(μ − A)H(λ) = H(λ)(I + dK(λ))(μ − A)

for λ sufficiently large. From these relations we obtain (μ − A)H(λ)(μ − A)−1 = (I − dL(λ))H(λ)(I + dK(λ)).


167

It is not difficult to show that the estimates (N2’) for H(λ) imply (N2’) for the similarity transform (μ−A)H(λ)(μ−A)−1 of H(λ) with ω replaced by some α, since K(t) and L(t) are exponentially bounded. Since μ − A induces an isomorphism between Y and X, (N2) for H(λ + α) follows. 2

6.5 Convergence of Resolvents Consider the sequence of problems t un (t) = fn (t) + An (t − τ )un (τ )dτ, 0

t ∈ J, n ∈ ⺞0 ,

(6.37)

where fn → f0 and An → A0 in an appropriate sense as n → ∞. Assuming existence of the pseudo-resolvents Sn (t) for (6.37) as well as the stability condition |Sn (t)|B(X) + sup {(t−s)−1 |Un (t)−Un (s)|B(Y ) } ≤ M eωt , 0<s 0, n ∈ ⺞0 , (6.38)

as in the case of the Trotter-Kato theorem on convergence of C0 -semigroups, the strong convergence Sn (t) → S0 (t) in X can be shown. Theorem 6.3 Let {An }n≥0 L1loc (⺢+ ; B(Y, X)) satisfy ∞⊂−ωt 1 (i) there is ϕ ∈ Lloc (⺢+ ), 0 e ϕ(t)dt < ∞ such that |An (t)y|X ≤ ϕ(t)|y|Y ,

for y ∈ Y, n ∈ ⺞, a.a. t > 0;

and (ii) there is a dense subset Z ⊂ Y such that An (t)z → A0 (t)z in X as n → ∞,

for z ∈ Z, a.a. t > 0.

Suppose Sn (t) are pseudo-resolvents for (6.37), n ∈ ⺞0 , such that the stability condition (6.38) holds. Then Sn (t)x → S0 (t)x as n → ∞, for each x ∈ X, t ≥ 0, and the convergence is uniform on compact subsets of ⺢+ × X. Proof: Without loss of generality, we may assume ω = 0. Next observe that (i) and (ii) imply An (t)y → A0 (t)y in X for a.a. t > 0, y ∈ Y as well as (i) also for n = 0. We therefore may set Z = Y . By virtue of (i), the Laplace transforms Aˆn (λ) of An (t), n ∈ ⺞0 , exist for λ ∈ ⺓+ , are holomorphic on ⺓+ with values in B(Y, X), and Aˆn (λ)y → Aˆ0 (λ)y in X as n → ∞, for each y ∈ Y and uniformly on right halfplanes Re λ ≥ η > 0. By the generation theorem, Theorem 6.2, I − Aˆn (λ) are invertible in X but also in Y and with Hn (λ) = Sˆn (λ) we have λHn (λ) = (I − Aˆn (λ))−1

in X

as well as |λHn (λ)|B(X) + |λHn (λ)|B(Y ) ≤ M,

λ > 0, n ∈ ⺞0 ,

(6.39)

168


note that the choice ε = 0 in Theorem 6.2 is possible, due to the stability condition (6.38). We show next Hn (λ)x → H0 (λ)x

as n → ∞ in X,

for all λ > 0, x ∈ X.

(6.40)

In fact, for each y ∈ Y we have (I − Aˆn (λ))−1 y − (I − Aˆ0 (λ))−1 y = (I − Aˆn (λ))−1 (Aˆn (λ) − Aˆ0 (λ))(I − Aˆ0 (λ))−1 y, hence Hn (λ)y − H0 (λ)y = λHn (λ)(Aˆn (λ) − Aˆ0 (λ))H0 (λ)y,

λ > 0, y ∈ Y.

Since Hn (λ)Y ⊂ Y for each n ∈ ⺞0 , λ > 0, by the strong convergence Aˆn (λ) → Aˆ0 (λ) and by (6.39) this identity implies (6.40) for each y ∈ Y, hence also for each x ∈ X, since Y is dense in X, and by (6.39) again. Observe that the convergence (6.40) is even uniform on compact subsets of (0, ∞), hence also uniform on compact subsets of (0, ∞) × X. Define operators Tn ∈ B(L1 (⺢+ ; X), X) by means of ∞ Tn f = Sn (τ )f (τ )dτ, f ∈ L1 (⺢+ ; X), n ∈ ⺞0 . (6.41) 0

If f is an exponential polynomial of the form f (t) =

N

e−λi t xi ,

t > 0,

(6.42)

i=1 N where {λi }N 1 ⊂ (0, ∞) and {xi }1 ⊂ X, then by (6.40)

Tn f =

N i=1

Hn (λi )xi →

N

H0 (λi )xi = T0 f,

as n → ∞.

i=1

The estimate |Tn f |X ≤ M |f |1 ,

f ∈ L1 (⺢+ ; X), n ∈ ⺞0 ,

implies uniform boundedness of the family {Tn }n≥0 ⊂ B(L1 (⺢+ ; X), X). In Lemma 6.2 below we show that exponential polynomials of the form (6.42) are dense in L1 (⺢+ ; X), hence by the Banach-Steinhaus theorem we obtain Tn f → T0 f as n → ∞ for each f ∈ L1 (⺢+ ; X). To prove the convergence of Sn (t), choose ft,n (s) = An (t − s)y · χ(0,t] ,

s > 0,

where y ∈ Y . By assumptions (i) and (ii) it is not difficult to show that the set {ft,n : 0 ≤ t ≤ τ, n ∈ ⺞0 } ⊂ L1 (⺢+ ; X) is relatively compact as long as y ∈ Y is fixed. Thus we obtain (Sn ∗ An y)(t) = Tn ft,n → T0 ft,0 = (S0 ∗ A0 y)(t)

as n → ∞,


169

uniformly for t ∈ [0, T ]. The second resolvent equations for (6.37) finally implies Sn (t)y → S0 (t)y as n → ∞ for each fixed y ∈ Y, uniformly for t ∈ [0, τ ], and the stability condition (6.38) then yields the strong convergence Sn (t)x → S0 (t)x for each x ∈ X, uniformly for t ∈ [0, τ ], by the Banach-Steinhaus theorem, since Y is dense in X. The proof is complete. 2

In the proof of Theorem 6.3 we used the following Lemma 6.2 Consider EP ⊂ L1 (⺢+ ; X) defined by EP = {f : f (t) =

N

e−λi t xi , for some N ∈ ⺞, λi > 0, xi ∈ X}.

i=1

Then EP is dense in L1 (⺢+ ; X). Proof: (a) We first derive a representation for functionals f ∗ ∈ (L1 (⺢+ ; X))∗ . Define x∗ (t) ∈ X ∗ by means of < x, x∗ (t) >X = < χ(0,t] x, f ∗ >,

x ∈ X, t ≥ 0;

obviously, we have x∗ (0) = 0, and for t > s |x∗ (t) − x∗ (s)| = sup | < χ(s,t] x, f ∗ > | ≤ |t − s||f ∗ |. |x|≤1

This shows x∗ ∈ Lip(⺢+ ; X ∗ ). Moreover, for a step function f ∈ L1 (⺢+ ; X) of the form N f (t) = χ(ti−1 ,ti ] (t)xi , t > 0, i=1

where 0 = t0 < t1 < · · · < tN < ∞, by linearity we obtain < f, f ∗ > =

N

< χ(ti−1 ,ti ] xi , f ∗ > =

i=1 ∞

=

N

< xi , x∗ (ti ) − x∗ (ti−1 ) >X

i=1

< f (t), dx∗ (t) > .

0

Since such step functions are dense in L1 (⺢+ ; X), by continuity we obtain the representation ∞ < f, f ∗ > = < f (t), dx∗ (t) > for all f ∈ L1 (⺢+ ; X). (6.43) 0

It is clear that the function x∗ (t) is uniquely determined by f ∗ , and by the convention x∗ (0) = 0. (b) Suppose EP ⊂ L1 (⺢+ ; X) is not dense. By the Hahn-Banach theorem, there is f ∗ ∈ (L1 (⺢+ ; X))∗ , f ∗ = 0, such that f ∗ ⊥ EP . Let x∗ ∈ Lip(⺢+ ; X ∗ )

170


denote the representation of f ∗ from (a). Then for all functions of the form f (t) = e−λt x, λ > 0, x ∈ X we obtain with (6.43) ∞ 0 = < f, f ∗ > = < f (t), dx∗ (t) > 0 ∞ ∞ −λt ∗ = < e x, dx (t) > = < x, e−λt dx∗ (t) >, 0

0

∗ (λ) = 0 for all λ > 0, which implies x∗ (t) ≡ x∗ (0) = 0; (6.43) then yields hence dx ∗ f = 0, a contradiction. Thus EP is dense in L1 (⺢+ ; X). 2

Observe that in Step (a) of this proof we have shown (L1 (⺢+ ; X))∗ ∼ = Lip0 (⺢+ ; X ∗ ) via the duality given by (6.43), where the subscript 0 refers to u(0) = 0. If X ∗ has the Radon-Nikodym property, then every Lipschitz-function x∗ ∈ Lip0 (⺢+ ; X ∗ ) has an a.e. derivative x˙ ∗ ∈ L∞ (⺢+ ; X ∗ ); hence in this case Lip0 (⺢+ ; X ∗ ) ∼ = L∞ (⺢+ ; X ∗ ), and the usual duality (L1 )∗ ∼ = L∞ applies again. Let us briefly specialize to the case of scalar equations with a fixed operator A, considered in Chapter 1, i.e. un (t) = fn (t) + (an ∗ Aun )(t),

t ∈ J, n ∈ ⺞0 .

(6.44)

In this situation, Theorem 6.3 can be strengthened and simplified as follows. Corollary }n≥0 ∈ L1loc (⺢+ ), A closed linear and densely defined, such ∞ 6.5 Let {an−ωt that 0 |an (t) − a(t)|e dt → 0 as n → ∞. Assume (6.44) admits a resolvent Sn (t) in X for each n ∈ ⺞ and that the stability condition |Sn (t)| ≤ M eωt ,

t ∈ ⺢+ , n ∈ ⺞,

(6.45)

holds. Then there is a resolvent S0 (t) of type (M, ω) for (6.44) with n = 0 and Sn (t)x → S0 (t)x as n → ∞, uniformly on compact subsets of ⺢+ × X. Observe that in contrast to the general case, existence of S0 (t) need not to be assumed, but can be proved. In fact, the convergence of an implies −1 (I − a → (I − a0 (λ)A)−1 n (λ)A)

in B(X)

for all Re λ > ω with aˆ0 (λ) = 0. Since points with aˆ0 (λ) = 0 are isolated in Re λ > ω, one then shows as in Section 1.5 that there are none. The stability condition yields (H1) and (H2) of the generation theorem for the scalar case, Theorem 1.3, and existence of S0 (t) follows. The asserted convergence finally is obtained as in the proof of Theorem 6.3. 6.6 Kernels of Positive Type in Hilbert spaces In the following, the spaces X and Y will always be Hilbert spaces.


171

Definition 6.4 Let A ∈ BVloc (⺢+ ; B(Y, X)) be of subexponential growth, i.e. ∞ e−εt |dA(t)|B(Y,X) < ∞ for each ε > 0. 0

The operator-valued kernel dA(t) is called of positive type, if

T

t

Re 0

0

(dA(τ )u(t − τ ), u(t))X dt ≥ 0

(6.46)

for each u ∈ C(⺢+ ; Y ) and T > 0. For instance, if A is a positive semidefinite selfadjoint operator in X, Y = XA , and a ∈ BVloc (⺢+ ) is such that da(t) is of positive type in the scalar sense (see Section 1.3), then dA(t) is of positive type in the sense of Definition 6.4, as we shall see below. As another example consider A(t) ≡ A where A is linear, densely defined and accretive; then dA(t) = δ(t)A is of positive type. Examples of nonscalar type will be given in Section 9, where kernels of positive type appear naturally. Kernels of positive type give rise to quite strong a priori estimates, the energy inequality. Proposition 6.6 Let A ∈ BVloc (⺢+ ; B(Y, X)), f ∈ W 1,1 (J; X), and suppose −dA(t) is of positive type. Then any mild solution of (6.1) verifies the energy inequality |u(t)|X ≤ |f (0)|X +

t

0

|f˙(τ )|X dτ,

t ∈ J.

(6.47)

In particular, mild solutions of (6.1) are unique. Proof: Since u ∈ C(J; X) is by assumption a mild solution of (6.1), there are (fn ) ⊂ C(J; X) and strong solutions (un ) ⊂ C(J; X) of (6.1) with f replaced by fn , such that fn → f and un → ∞u in C(J; X) as n → ∞. Choose mollifiers ρε ∈ C0∞ (0; ε) with ρε ≥ 0 and −∞ ρε (τ )dτ = 1, and define fnε = fn ∗ ρε , unε = un ∗ ρε , fε = f ∗ ρε , uε = u ∗ ρε . Then fnε ∈ C 1 (J; X), unε ∈ C 1 (J; Y ) and we have u˙ nε = dA ∗ unε + f˙nε , unε (0) = fnε (0) = 0. From this we obtain d |unε (t)|2 = 2Re(u˙ nε (t), unε (t)) = 2Re(dA ∗ unε (t), unε (t)) + 2Re(f˙nε (t), unε (t)), dt hence after integration and by (6.46) t t |unε (t)|2 ≤ 2Re (dA ∗ unε (s), unε (s))ds + 2 |f˙nε (s)||unε (s)|ds 0 0 t ≤ 2 |f˙nε (τ )||unε (τ )|dτ, t ∈ J. 0

172


Denote the right hand side of this inequality by ψ(t); then ˙ ψ(t) 2|f˙nε (t)||unε (t)| d ψ(t) = = ≤ |f˙nε (t)| dt 2 ψ(t) 2 ψ(t) for each t ∈ J, such that ψ(t) > 0. Therefore we conclude |unε (t)| ≤

ψ(t) ≤

t 0

|f˙nε (τ )|dτ,

t ∈ J.

Letting first n → ∞ we obtain |uε (t)| ≤

t 0

t t |f˙ε (τ )|dτ ≤ ( ρε (τ )dτ )|f (0)| + (ρε ∗ |f˙|)(τ )dτ 0

0

and then with ε → 0+ Inequality (6.47) follows. 2

As in the scalar case, operator-valued kernels can be characterized in terms of their Laplace transforms. This characterization is useful in theory but even more in applications. Proposition 6.7 Let A ∈ BVloc (⺢+ ; B(Y, X)) be of subexponential growth. Then dA(t) is of positive type if and only if Re (dA(λ)y, y)X ≥ 0

for each

y ∈ Y, Re λ > 0.

(6.48)

Proof: (⇒) Suppose dA(t) is of positive type, and let y ∈ Y, λ ∈ ⺓+ be given. Choosing u(t) = e−λt y, t > 0, (6.46) yields

T

t

Re 0

0

(dA(τ )e−λ(t−τ ) y, e−λt y)dt ≥ 0,

for each T > 0;

since A(t) is of subexponential growth, the limit as T → ∞ of the left hand side exists. From this we obtain ∞ t 0 ≤ Re (dA(τ )e−λ(t−τ ) y, e−λt y)dt 0 0 ∞ ∞ = Re ( e−(λ+λ)t dt)eλτ (dA(τ )y, y) 0 τ ∞ 1 1 = Re e−(λ+λ)τ eλτ (dA(τ )y, y) = Re(dA(λ)y, y), 2Reλ 2Reλ 0 and so (6.48) follows.


173

(⇐) Conversely, assume (6.48) holds and let u ∈ C(J; Y ) be given. Extending u(t) by zero to all of ⺢ and using Parseval’s identity in the Hilbert space L2 (⺢; X) we obtain for any μ > 0

T

t

e−μτ (dA(τ )u(t − τ ), u(t))dt

2πRe

0

0

−∞ ∞

∞

= Re = Re −∞

t ([ dA(τ )e−μτ u(t − τ )](ρ), u˜(ρ))dρ 0

+ iρ)˜ (dA(μ u(ρ), u ˜(ρ))dρ ≥ 0,

and with μ → 0+, (6.46) follows. 2

As a consequence of Proposition 6.7, if −dA(t) is of positive type then the operators dA(λ), Re λ > 0, with domain Y are dissipative in X; in particular they are closable, and the inequality Re λ|y|X ≤ |(λ − dA(λ))y| X,

y ∈ Y, Re λ > 0

is satisfied. If in addition the range of λ − dA(λ) is dense in X, then H(λ) = −1 −1 = (λ − dA(λ)) (λ − dA(λ)) is well-defined and we have the estimate |H(λ)| ≤ 1/Re λ. If in addition dA(λ) with domain Y is closed in X for each λ > 0, or equivalently λ − dA(λ) is surjective for each λ > 0 then the following result holds. Theorem 6.4 Let A ∈ BVloc (⺢+ ; B(Y, X)) be of subexponential growth, suppose that −dA(t) is of positive type and that λ − dA(λ) is surjective for each λ > 0. Then −1 (i) H(λ) = (λ − dA(λ)) satisfies |(

d n ) H(λ)|B(X) ≤ n!λ−(n+1) , dλ

for all λ > 0, n ∈ ⺞0 .

(6.49)

(ii) There is a unique weak resolvent S(t) for (6.1), i.e. a family {S(t)}t≥0 ⊂ B(X) satisfying (S1) and the second resolvent equation (6.3). 1,∞ (iii) |S(t)|B(X) ≤ 1 for each t ≥ 0, and S(·)y ∈ Wloc (⺢+ ; X) for each y ∈ Y. Proof: (i) Define operators A0n in X with domain D(A0n ) = Y by means of A0n = dA(n), n ∈ ⺞. Then A0n is m-dissipative, hence generates a C0 -semigroup of contractions in X; note that Y = XA0n for each n ∈ ⺞. Next we define Bn ∈ L1loc (⺢+ ; B(Y, X)) by means of Bn (t) =

∞ n2k k=1

k!

dA

(k)

(n)(−1)k

tk−1 −nt e , (k − 1)!

t > 0, n ∈ ⺞;

(6.50)

174


a simple estimate yields 0

∞

e−εt |Bn (t)|B(Y,X) dt ≤

∞ n2k k=1

k!

|dA

(k)

(n)|B(Y,X) · (n + ε)−k

∞ ∞ n2k ≤ ( tk e−nt |dA(t)|B(Y,X) )(n + ε)−k k! 0 k=0 ∞ ∞ ∞ 2 nεt 1 n t k −nt = ( ( ) )e |dA(t)|B(Y,X) = e− n+ε |dA(t)|B(Y,X) < ∞, k! n + ε 0 0 k=0

for each ε > 0, n ∈ ⺞, showing that Bn is of subexponential growth. A straightforward computation gives λn ) − dA(n), ˆn (λ) = dA( B λ+n

λ > 0, n ∈ ⺞.

(6.51)

1,1 Differentiating (6.50) we see that Bn ∈ Wloc (⺢+ ; B(Y, X)) and ∞ ∞ nεt e−εt |B˙ n (t)|B(Y,X) dt ≤ (2n + ε) e− n+ε |dA(t)|B(Y,X) < ∞ 0

0

for each ε > 0, n ∈ ⺞. By Corollary 6.2, there is a resolvent Sn (t) for the problem u˙ = A0n u + Bn ∗ u + f,

u(0) = x,

1,1 since A0n generates a C0 -semigroup in X and Bn ∈ Wloc (⺢+ ; B(Y, X)). Let t 0 ˆ ˆ n (λ))/λ = An (t) be defined by An (t) = An + 0 Bn (s)ds; then An (λ) = (A0n + B λn dA( λ+n )/λ, by (6.51), and so

n (λ)y, y) = Re (dA( Re (dA

λn )y, y) ≤ 0 λ+n

for each y ∈ Y, Re λ > 0;

λn observe that Re λ+n > 0 for each Re λ > 0, n ∈ ⺞. By Proposition 6.7, −dAn (t) = −A0n δ(t) − Bn (t)dt is of positive type, and so from Proposition 6.6 we obtain the uniform bound |Sn (t)|B(X) ≤ 1, t ≥ 0, n ∈ ⺞. Taking Laplace transforms this gives

|(

d mˆ ) Sn (λ)|B(X) ≤ m!( Re λ)−(m+1) dλ

for all

Re λ > 0, m, n ∈ ⺞.

(6.52)

ˆ λn ))−1 /λ, λ > 0, n ∈ ⺞, Theorem 6.2 yields Sˆn (λ) = (I − Aˆn (λ))−1 /λ = (I − A( λ+n and therefore Sˆn (λ) → H(λ) as n → ∞, for each λ > 0. From the holomorphy of Sˆn (λ) on Re λ > 0 and (6.52) we now conclude that (6.49) is valid. (ii) Theorem 0.2 and (6.49) imply the existence of a Lipschitz family {U (t)}t≥0 ⊂ ˆ (λ) = H(λ)/λ, λ > 0 holds. The definition of U (t) yields the B(X) such that U


175

second resolvent equation in integrated form (6.6). Since A ∈ BVloc (⺢+ ; B(Y, X)) by assumption, U (t)x is C 1 for each x ∈ Y, hence even for each x ∈ X, by the Banach-Steinhaus Theorem. Thus S(t) = U˙ (t) satisfies (S1) and (6.6), hence also (6.3) by differentiation, i.e. S(t) is a weak resolvent. (iii) The first assertion is obvious, and the second follows from the second resolvent equation. 2

It is clear that for weak resolvents S(t) the variation of parameters formula (6.7) is still valid, since in its derivation at the beginning of Section 6.2 only (S1) and the second resolvent equation were used. In particular, mild solutions are then unique. On the other hand, (6.10) makes sense for such S(t) if f ∈ W 1,1 (J; X), however, it is not clear that the resulting function u ∈ C(J; X) is a mild solution, even if f is much better, say f ∈ W 1,1 (J; Y ). The reason for this is of course that S(t) or even U (t) cannot be expected to leave the space Y invariant, unless some further conditions are satisfied. We mention two such conditions in the following corollaries. Corollary 6.6 In the situation of Theorem 6.4, assume in addition that A(t) has a main part, i.e. is of the form (6.35) of Corollary 6.4. Then the weak resolvent S(t) of Theorem 6.4 is an a-regular resolvent for (6.1). Proof: Combine Theorem 6.4 with Corollary 6.4 and Proposition 6.2; observe that X has the Radon-Nikodym property since it is even a Hilbert space. 2

Corollary 6.7 In the situation of Theorem 6.4, assume in addition that there is d

a third Hilbert space Z → Y, such that the restriction Az (t) of A(t) to Z fulfills the assumptions of Theorem 6.4 with X and Y replaced by Y and Z, respectively. Then the weak resolvent S(t) of Theorem 6.4 is a resolvent in X. Proof: By Theorem 6.4, H(λ) verifies (6.30), hence there is a pseudo-resolvent S(t) in X for (6.1) by Theorem 6.2, which is already a resolvent by Proposition 6.2. By uniqueness, S(t) coincides with the weak resolvent from Theorem 6.4 in X and also in Y. 2

6.7 Hyperbolic Problems of Variational Type d

Let V and H denote Hilbert spaces such that V → H, and let ((·, ·)) resp. (·, ·) denote the inner products in V resp. H, and || · || resp. | · | the corresponding ∗

d

∗

norms. Identifying the antidual H of H with H, by duality we also get H → V , ∗ where V means the antidual of V . The relation between the antiduality < ·, · > ∗ between V and V and the inner product in H is given by (v, h) = < v, h >, The norm in V

∗

will be denoted by || · ||∗ .

v ∈ V, h ∈ H.

176


Let α : ⺢+ × V × V → ⺓ be such that α(t, ·, ·) is a bounded sesquilinear form on V for each t ≥ 0, and |α(t, u, v) − α(s, u, v)| ≤ (α0 (t) − α0 (s))||u|| ||v||,

for all t ≥ s ≥ 0, u, v ∈ V, (6.53) ∗ where α0 is nondecreasing. Suppose f ∈ C(J; V ) is given and consider the Volterra equation of variational type t (w, u(t)) + α(t − s, w, u(s))ds = < w, f (t) >, t ∈ J, w ∈ V. (6.54) 0

∗

Representing the forms α(t, ·, ·) by bounded linear operators A(t) ∈ B(V, V ) as α(t, w, u) = − < w, A(t)u >,

t ≥ 0, w, u ∈ V,

(6.55)

∗

it is clear that (6.54) can be written as (6.1) in X = V with Y = V ; observe ∗ A ∈ BVloc (⺢+ ; B(V, V )) by (6.53). Exploiting the variational structure of (6.54), by the methods introduced in ∗ Section 6.6 it is possible to prove existence of the resolvent S(t) of (6.54) in V and to derive a number of further properties of S(t), provided the form α is coercive in the sense of the following definition. Definition 6.5 A form α : ⺢+ × V × V → ⺓ as above is called coercive if there is a constant γ > 0 such that

T

2 Re 0

t ( dα(s, u(t), u(t − s)))dt ≥ γ|| 0

T

u(t)dt||2 ,

(6.56)

0

for all u ∈ C(⺢+ ; V ) and T > 0. ∗

Suppose u ∈ C(J; V ) is a solution of (6.54) and let f ∈ W 1,1 (J; V ), f (0) ∈ H. Similarly to Proposition 6.6, the coercive estimate (6.56) then yields the inequality |u(t)|2 + γ||

t 0

u(τ )dτ ||2 ≤ |f (0)|2 + 2 Re

t

< u(τ ), f˙(τ ) > dτ,

(6.57)

0

for all t ≥ 0. This is the basic energy inequality for (6.54) in the case of coercive forms. If α is only positive, i.e. γ = 0, (6.57) is still valid; however, we then do not obtain bounds on any quantity related to the solution u(t) in V . It turns out that (6.57) implies estimates for mild solutions as well. ∗

Proposition 6.8 Suppose u ∈ C(⺢+ ; V ) is a mild solution of (6.54). Then 1,1 (i) f ∈ Wloc (⺢+ ; H) implies u ∈ C(⺢+ ; H), 1 ∗ u ∈ C(⺢+ ; V ), and |u(t)|2 + γ||

t 0

u(τ )dτ ||2 ≤ (|f (0)| +

t 0

|f˙(τ )|dτ )2 ,

t ≥ 0;

(6.58)


177

∗

2,1 (ii) f ∈ Wloc (⺢+ ; V ), f (0) = 0, imply u ∈ C(⺢+ ; H), 1 ∗ u ∈ C(⺢+ ; V ), and for each δ ∈ (0, γ)

|u(t)|2 + (γ − δ)||

t 0

u(τ )dτ ||2 ≤ c(γ, δ)2 (||f˙(0)||∗ +

t 0

|f¨(t)||∗ dτ )2 ,

(6.59)

for all t ≥ 0, where c(γ, δ) = γ 1/2 + (γ − δ)−1/2 . ∗

Proof: Since u ∈ C(⺢+ ; V ) is by assumption a mild solution of (6.54), there are ∗ (fn ) ⊂ C(⺢+ ; V ) and strong solutions (un ) ⊂ C(⺢+ ; V ) of (6.54) with f replaced ∗ by fn , such that fn → f and un → u in C(⺢+ ; V ) as n → ∞. Choose mollifiers ∞ ρε ∈ C0∞ (0, ε) with ρε ≥ 0 and −∞ ρε (τ )dτ = 1, and define fnε = fn ∗ ρε , ∗

unε = un ∗ ρε , fε = f ∗ ρε , uε = u ∗ ρε . Then fnε ∈ C 1 (⺢+ ; V ), unε ∈ C 1 (⺢+ ; V ) and we have from (6.57)

2

|unε (t)| + γ||

t 0

2

unε (τ )dτ || ≤ 2Re

t

0

˙ (τ ) > dτ, < unε (τ ), fnε

(6.60)

for all t ≥ 0, since fnε (0) = 0. t (i) Let ψ(t) = 2 0 |fnε (τ )||unε (τ )|dτ for t > 0; then ˙ d ψ(t) 2|f˙nε (t)||unε (t)| ψ(t) = = ≤ |f˙nε (t)|, dt 2 ψ(t) 2 ψ(t) for each t ∈ ⺢+ , such that ψ(t) > 0. Therefore we conclude (|unε (t)|2 + γ|

t 0

unε (τ )dτ |2 )1/2 ≤

ψ(t) ≤

t 0

|f˙nε (τ )|dτ,

t ∈ ⺢+ .

Letting first n → ∞ we obtain 2

|uε (t)| + γ||

t 0

t t 2 ˙ uε (τ )dτ || ≤ ( |fε (τ )|dτ ) ≤ ( (ρε ∗ |df |)(τ )dτ )2 , 2

0

0

and then with ε → 0+, inequality (6.58) follows. (ii) For the proof of (ii), let w(t) = 1 ∗ u(t), wn (t) = 1 ∗ un (t), and δ ∈ (0, γ); we integrate by parts in (6.60), rearrange, and estimate. |unε (t)|2 + γ||wnε (t)||2 ≤ 2Re

t

0 t

˙ (τ ) > dτ < unε (τ ), fnε

= 2Re < wnε (t), f˙nε (t) > −2Re < wnε (τ ), f¨nε (τ ) > dτ 0 t ˙ (t)||2 + 2 ≤ δ||wnε (t)||2 + δ −1 ||fnε ||wnε (τ )|| · ||f¨nε (τ )||∗ dτ. ∗ 0

178


Setting ˙ (t)||2 + 2 ψ(t) = δ −1 ||fnε ∗

t 0

||wnε (τ )|| · ||f¨nε (τ )||∗ dτ,

we obtain ˙ ψ(t) ≤ 2δ −1 ||f˙nε (t)||∗ · ||f¨nε (t)||∗ + 2||wnε (t)|| · ||f¨nε (t)||∗ ≤ 2||f¨nε (t)||∗ · (δ −1/2 + (γ − δ)−1/2 ) ψ(t), hence we conclude as in (i) t t 2 2 1/2 −1/2 −1/2 (|unε (t)| + (γ − δ)|| unε (τ )dτ || ) ≤ (δ + (γ − δ) ) ||f¨nε (τ )||∗ dτ, 0

0

for all t ≥ 0. Letting first n → ∞ and then ε → 0, assertion (ii) follows. 2

Since α0 is assumed to be of subexponential growth, coerciveness can be characterized in terms of Laplace transforms; similar to kernels of positive type. For the sake of completeness, we include an indication of proof, although it is similar to that of Proposition 6.7. Proposition 6.9 Let α(t; ·, ·) be a sesquilinear form on V such that (6.53) is satisfied, where α0 is nondecreasing and of subexponential growth. Then α satisfies (6.56) iff u, u) ≥ γRe(1/λ)||u||2 , Re dα(λ;

for each u ∈ V, and Re λ > 0.

(6.61)

Proof: Replacing α(t; ·, ·) by α(t; ·, ·) − (γt/2)((·, ·)), it is sufficient to verify the equivalence of (6.56) and (6.61) for γ = 0. (⇒) Suppose α is of positive type, and let u ∈ V, λ ∈ ⺓+ be given. Choosing u(t) = e−λt u, t > 0, (6.56) yields

T

t

2 Re 0

0

dα(τ ; e−λt u, e−λ(t−τ ) u)dt ≥ 0,

for each T > 0;

since α0 (t) is of subexponential growth, the limit as T → ∞ of the left hand side exists. From this we obtain for T → ∞ ∞ t 0 ≤ 2Re dα(τ ; e−λt u, e−λ(t−τ ) u)dt 0 ∞ 0 ∞ = 2Re ( e−(λ+λ)t dt)eλτ dα(τ ; u, u) 0 τ ∞ 1 1 Re e−(λ+λ)τ eλτ dα(τ ; u, u) = Re dα(λ; u, u), = Re λ Re λ 0 and so (6.61) with γ = 0 follows.


179

(⇐) Conversely, assume (6.61) holds and let u ∈ C(⺢+ ; V ) be given. Redefining u(t) = 0 for t ∈ [0, T ] and using the Fourier inversion formula and Fubini’s theorem, we obtain for any μ > 0 T t 2πRe e−μτ dα(τ ; u(t), u(t − τ ))dt 0 0 ∞ t ˜˜(−t), = Re dt −∞ 0 ∞ + iρ)˜ = Re dρ −∞ ∞ − iρ; u = Re dα(μ ˜(ρ), u ˜(ρ))dρ ≥ 0, −∞

and with μ → 0+, (6.56) with γ = 0 follows. 2

Examples for coercive forms in linear thermoviscoelaticity will be given in Section 9.4. The result announced above reads as follows. Theorem 6.5 Suppose α : ⺢+ × V × V → ⺓ satisfies (V1) α(t, ·, ·) is a bounded sesquilinear form on V, for t ≥ 0, and α(0, ·, ·) = 0; 1,∞ (V2) α(·, u, v) ∈ Wloc (⺢+ ) for each u, v ∈ V, and |α(t, ˙ u, v) − α(s, ˙ u, v)| ≤ (α1 (t) − α1 (s))||u|| ||v||,

u, v ∈ V, t ≥ s ≥ 0,

where α1 (t) is nondecreasing and of subexponential growth, α1 (0) = 0; (V3) α is coercive with coercivity constant γ > 0. ∗ ∗ 1,∞ Define A ∈ Wloc (⺢+ ; B(V, V )) by means of (6.55). Then for X = V and Y = V, (6.1) admits a t-regular resolvent S(t). Moreover, with R = 1 ∗ S and T = t ∗ S we have in addition ∗ ∗ (a) {S(t)}t≥0 ⊂ B(V ) ∩ B(V ) ∩ B(H) is strongly continuous in V , H, and V, and |S(t)|B(V ) , |S(t)|B(V ∗ ) ≤ 1 + 2γ −1 α1 (t), |S(t)|B(H) ≤ 1, for all t ≥ 0; ∗

∗

(b) {R(t)}t≥0 ⊂ B(V , H) ∩ B(H, V ), {T (t)}t≥0 ⊂ B(V , V ) are strongly continuous, and |R(t)|B(V ∗ ,H) , |R(t)|B(H,V ) ≤ γ −1/2 , |T (t)|B(V ∗ ,V ) ≤ 2γ −1 , for all t ≥ 0; ∗

(c) {S(t)}t≥0 ⊂ B(V, H) ∩ B(H, V ) is strongly continuously differentiable, even ∗ twice a.e. in B(V, V ), and for a.a. t ≥ 0 we have −1/2 ˙ ˙ |S(t)| α1 (t), B(V,H) , |S(t)|B(H,V ∗ ) ≤ γ

and

−1 ¨ ∗ |S(t)| α1 (t)). B(V ;V ) ≤ α1 (t)(1 + 2γ

180


Before we turn to the proof of Theorem 6.5 several remarks are in order. As∗ ∗ sumption (V2) means A˙ ∈ BVloc (⺢+ ; B(V, V )), even in BV (⺢+ ; B(V, V )) if α1 is bounded. Therefore (6.54) is equivalent to the equation of second order

t

v¨(t) = 0

˙ )v(t − τ ) + g(t), dA(τ

v(0) = v0 ,

t ≥ 0,

(6.62)

v(0) ˙ = v1 .

For the solution of (6.62) we have the following variation of parameters formula. v(t) = S(t)v0 + R(t)v1 +

t 0

R(t − τ )g(τ )dτ,

t ≥ 0.

(6.63)

Thus the resolvent S(t) corresponds to the cosine family, R(t) to the sine family of second order differential equations. The following corollary describes the solvability behaviour of (6.62) implied by Theorem 6.5. ∗

Corollary 6.8 Let the assumptions of Theorem 6.5 be satisfied, v0 , v1 ∈ V , ∗ g ∈ L1loc (⺢+ ; V ), and let v(t) be given by (6.63). Then ˙ (i) v0 ∈ V, v1 ∈ H, g ∈ C(⺢+ ; H) imply v ∈ C(⺢+ ; V ), v˙ ∈ C(⺢+ ; H), v¨ − A(·)v 0 ∈ ∗ C(⺢+ ; V ), and v(t) is a strong solution of (6.62); ∗ (ii) v0 ∈ V, v1 ∈ H, g ∈ W 1,1 (⺢+ ; V ) imply v ∈ C(⺢+ ; V ), v˙ ∈ C(⺢+ ; H), ∗ ˙ v¨ − A(·)v 0 ∈ C(⺢+ ; V ), and v(t) is a strong solution of (6.62); ∗ ∗ ∗ (iii) v0 ∈ H, v1 ∈ V , g ∈ C(⺢+ ; V ) imply v ∈ C(⺢+ ; H), v˙ ∈ C(⺢+ ; V ), and v(t) is a mild solution of (6.62). Proof: The construction is similar to the proof of Theorem 6.4, therefore the exposition will be kept brief, here. Introduce sesquilinear forms αn by means of ˙ α˙ n (t, u, v) = α(n, u, v) +

∞ n2k k=1

k−1 ˙ (k) (n, u, v)(−1)k t α e−nt , k! (k − 1)!

for t ≥ 0, u, v ∈ V, n ∈ ⺞; these forms αn are obviously bounded and sesquilinear; they are exactly those forms corresponding to the operator An (t) in the proof of Theorem 6.4, and so the approximating equations with A(t) replaced by An (t) ∗ admit resolvents Sn (t) in V . Since n (λ, u, v) = dα( λn , u, v), dα λ+n

Re λ > 0, n ∈ ⺞, u, v ∈ V,

αn is coercive with the same constant γ of coercivity. A priori estimates (6.58) ∗ and (6.59) yield for f (t) ≡ h ∈ H resp. f (t) = tv ∗ ∈ V |Sn (t)|B(H) ≤ 1, |Rn (t)|B(H,V ) , |Rn (t)|B(V ∗ ,H) ≤ γ −1/2 , |Tn (t)|B(V ∗ ,V ) ≤ 2γ −1 ,


181

for all t > 0, where Rn = 1 ∗ Sn , Tn = t ∗ Sn . These inequalities imply d kˆ ) Sn (λ)|B(H) ≤ k!λ−(k+1) , dλ d |( )k [Sˆn (λ)/λ]|B(V ∗ ,H) ≤ γ −1/2 k!λ−(k+1) , dλ d |( )k [Sˆn (λ)/λ]|B(H,V ∗ ) ≤ γ −1/2 k!λ−(k+1) , dλ d |( )k [Sˆn (λ)/λ2 ]|B(V ∗ ,V ) ≤ 2γ −1 k!λ−(k+1) , dλ

|(

for all n ≥ 1, k ≥ 0, λ > 0. Hence passing to the limit n → ∞ the same ∗ ˙ −1 estimates are valid for H(λ) = (λ − A(λ)) ; note that {H(λ)}Re λ>0 ⊂ B(V , V ) exists and is holomorphic, as standard coerciveness arguments show, and that ∗ (k) Sˆn (λ) → H (k) (λ) in B(V , V ) uniformly on compact subsets of ⺓+ , and for all k. The identities 2 ˙ ˙ H(λ) = I/λ + d A(λ)H(λ)/λ = I/λ + [H(λ)/λ2 ]d A(λ),

λ > 0,

by (V2) imply for each ε > 0 |(

d k d 1 (ε))k!(λ − ε)−k−1 , ) H(λ)|B(V ∗ ) , |( )k H(λ)|B(V ) ≤ (1 + 2γ −1 dα dλ dλ

this follows from the estimate for H(λ)/λ2 and from ∞ ∞ (k) 1 (ε), ˙ (λ − ε)k |d A˙ (λ)|/k! ≤ e−εt |dA(t)| ≤ dα

λ > ε;

ε > 0.

0

k=0

Therefore, the assumptions of the generation theorem, Theorem 6.2, are satisfied ∗ in X = V with Y = V, hence with a(t) = t, there is a t-regular resolvent S(t) for (6.1). The remaining assertions follow easily from the a priori estimates (6.58) ∗ and (6.59) and from the resolvent equations in V . 2

To compare Theorem 6.5 with Corollary 1.2, let α(t, u, v) = a(t)(u, Av) for some positive definite selfadjoint operator A in H, V = D(A1/2 ) equipped with the inner t product ((u, v)) = (A1/2 u, A1/2 v), and a(t) = a∞ t + 0 a1 (τ )dτ , where a∞ ≥ 0, and a1 of positive type. Then α is coercive if γ = a∞ > 0, and Theorem 6.5 applies if in additon a1 ∈ BVloc (⺢+ ) and α1 (t) = Var a1 |t0 is of subexponential growth. Another coercivity concept different from Definition 6.5 is based on an inequality of the form v, v) ≥ γRe Re dα(λ,

1 ||v||2 , λ+η

v ∈ V, Re λ > 0,

(6.64)

where γ and η are positive constants; compare with (6.61). Forms satisfying (6.64) will be called η-coercive in the sequel. By the same methods as in the proof of Theorem 6.5 the following result for η-coercive forms is obtained.

182


Corollary 6.9 Suppose α : ⺢+ × V × V → ⺓ is η-coercive for some η > 0, and satisfies (V1), (V2) of Theorem 6.5. Let A(t) be defined by (6.55) again. ∗ Then for X = V , Y = V, (6.1) admits a t-regular resolvent S(t). Moreover, with Rη (t) = (e−ηt ∗ S)(t), Tη (t) = (e−ηt ∗ Rη )(t) we have the estimates |S(t)|B(H) ≤ 1; and

|Rη (t)|B(V ∗ ,H) , |Rη (t)|B(H,V ) ≤ γ −1/2 ;

||Rη (·)x||2 ≤ (2ηγ)−1/2 |x|;

|Tη (t)|B(V ∗ ,V ) ≤ 2γ −1 ,

||Tη (·)x||2 ≤ 2(2ηγ 2 )−1/2 ||x||∗ .

In the situation of Corollary 6.9 one can also obtain bounds for the remaining ∗ ∗ ˙ quantities, i.e. S(t) in B(V ) and B(V ), S(t) in B(V, H) and B(H, V ), as well as ∗ ¨ in B(V, V ), to the result that similar estimates as in Theorem 6.5 are valid; S(t) in particular, all of these operator families are bounded on ⺢+ if α1 is bounded. Concerning the proof of Corollary 6.9 we only mention the basic energy inequality corresponding to (6.58) which is implied by η-coercivity. t t 2 2 2 2 |v(t)| + γ||w(t)|| + 2ηγ ||w(s)|| ds ≤ |f (0)| + 2 Re < v(s), f˙(s) > ds, 0

0

(6.65) for all t ≥ 0, where w(t) = e−ηt ∗ v(t). 6.8 Comments d a) The setting for (6.1), i.e. Y → X, A ∈ L1loc (⺢+ ; B(Y ; X)), is fairly general and seems to cover all abstract treatments of Volterra equations which have been considered so far. Of course, this is not the most general approach one can think of, e.g. the domains D(A(t)) of the operators A(t) could be imagined to vary with t in a much stronger way. However, then it becomes very difficult to develop a reasonable theory. From the applications point of view, our setting covers most of the equations arising in linear viscoelasticity, at least for the variational approach. b) By far the most papers in the literature deal with first order equations of the form u˙ = Au + B ∗ u + f, u(0) = x, (6.66) where A generates a C0 -semigroup in X or even an analytic semigroup, and B(t) is ‘dominated’ by A in the sense that at least B(·)x is measurable in X for each x ∈ D(A), and |B(t)x| ≤ ϕ(t)(|x| + |Ax|) a.e. for some ϕ ∈ L1loc (⺢+ ); see e.g. Chen and Grimmer [40], [41], Grimmer [139], Grimmer and Kappel [140], Grimmer and Pritchard [144], Grimmer and Schappacher [146], Miller [243], Miller and Wheeler [245], Da Prato and Iannelli [65], Tsuruta [329], and many others. Further references are given below. Grimmer and Pr¨ uss [145] contains the first general treatment of (6.66), i.e. the connection between well-posedness and resolvents, the generation theorem, the 1-regularity of the resolvent and several counterexamples, some of which have been mentioned before. Observe that (6.66) is of the form (6.12) with a(t) ≡ 1, hence Propositions 6.2, 6.4 and 6.5(c) as well as Corollaries 6.2 and 6.4 are applicable.


183

c) Although Corollary 6.2 with B0 = 0 has many predecessors, its present form is originally due to Desch and Schappacher [93], who used the semigroup approach via history spaces for the proof. Grimmer and Pr¨ uss [145] gave an independent argument based on operator-valued bounded Volterra equations. This approach was later improved and extended in Pr¨ uss [276], and by Pugliese [286]. Theorem 6.1 and its corollaries unify most results of this nature, although in special situations better results are known. d) The scalar version of the generation theorem, i.e. Theorem 1.3 for the case a ∈ BVloc (⺢+ ), is due to Da Prato and Iannelli [65]. Grimmer and Pr¨ uss [145] generalized their result to first order equations of the form (6.66), while Pugliese [286] contains a version of Theorem 6.2 for the case A ∈ BVloc (⺢+ ; B(Y, X)); see also Sforza [300]. The simple but nevertheless interesting Example 6.2 is a slightly modified version of an example due to Desch and Schappacher [93]. e) A second class of problems of the form (6.1) which has been considered by many authors is the class of second order equations u ¨ = Au + B ∗ u + f, u(0) = x, u(0) ˙ = y,

(6.67)

where A is the generator of a strongly continuous cosine family in X, and the operator family B(t) belongs to BVloc (⺢+ ; B(XA , X)); see e.g. Adali [2], Dafermos [76], [75], Desch and Grimmer [82], and Tsuruta [330]. The results in the literature as well as Corollary 6.3 for (6.54) are not optimal for the complete second order differential equation u ¨ = Au + B u˙ + f, u(0) = x, u(0) ˙ = y; see Clément and Pr¨ uss [53] Fattorini [114], Obrecht [263], [264] and Watanabe [335] for stronger results. f) The convergence theorem, Theorem 6.3, seems to be new. The proof follows an idea of F. Neubrander, who proved a general convergence theorem for vector-valued Laplace transforms. It would be interesting to have a version of this result which does not assume the existence of the limiting pseudo-resolvent, and only involves the convergence of the Laplace-transforms of the approximating resolvents, not of the kernels itself, similar to the classical Trotter-Kato theorem; see e.g. Goldstein [133] for the latter. Other versions of Corollary 6.5, in which the kernel a(t) is fixed but A depends on n ∈ ⺞0 , can be found in Lizama [216]. g) It is of course very well known that the duality (6.43) yields L1 (⺢+ ; X)∗ ∼ = Lip0 (⺢+ ; X); see Diestel and Uhl [99] and Hashimoto and Oharu [173]. h) The results in Section 6.6 on operator-valued kernels of positive type are different from but similar to those obtained by Da Prato and Iannelli, [65]; see also Barbu, Da Prato and Iannelli [18] for results on asymptotic behaviour in

184


the setting of the paper just cited. An interesting special case of such kernels is A(t) = a(t)A + b(t)B, where a, b are scalar kernels such that da, db are of positive type and A, B are selfadjoint and positive semidefinite. Even in this situation the regularity of the weak resolvent S(t) is unknown except if A, B commute. i) A theory for nonautonomous linear problems has been developed for first order equations of the form t u(t) ˙ = A(t)u(t) + B(t, τ )u(τ )dτ, u(s) = x, t ≥ s, (6.68) s

where the operator family A(t) generates an evolution operator in X, and B(t, s) is subordinate to A(t) in a suitable sense; see Chen and Grimmer [40], [41], Grimmer [139], and Pr¨ uss [270] for the hyperbolic case, and Acquistapache [1], Di Blasio [95], Friedman and Shinbrot [123], Lunardi [225], Pr¨ uss [270], and Tanabe [320], [321] in parabolic situations. In all of these papers the idea is to consider the integral term as a perturbation of the evolution equation u(t) ˙ = A(t)u(t) + f (t) and to use the variation of parameters formula. This is similar to the approach taken in Sections 6.3 for the hyperbolic case, and to that of Sections 7.4, 7.5, 7.6 as well as of Sections 8.5 and 8.6 below in the parabolic case.

7

Nonscalar Parabolic Equations

The results on existence of analytic resolvents in Section 2 and on resolvents for parabolic problems in Section 3 as well as the corresponding results on maximal regularity of type C α are here extended to nonscalar equations. Particularly easy to verify are the conditions in the variational approach in Section 7.3 which continues the discussion begun in Section 6.7. The remaining subsections are devoted to a far reaching improvement of the perturbation theorem from Section 6.3 in the parabolic case. 7.1 Analytic Resolvents Consider again (6.1) in a Banach space X, i.e. t u(t) = f (t) + A(t − τ )u(τ )dτ, 0

t ∈ J,

(7.1)

d

where Y → X, A ∈ L1loc (⺢+ ; B(Y, X)), f ∈ C(J; X), and J = [0, T ]. We begin the study of (7.1) in the parabolic case with an extension of Theorem 2.1 on analytic resolvents for scalar equations to (7.1). In analogy to Definition 2.1 we introduce Definition 7.1 (i) A weak resolvent S(t) for (7.1) is called analytic if the function S(·) : ⺢+ → B(X) admits analytic extension to a sector Σ(0, θ0 ), θ0 ≤ π/2. Then S(t) will be said to be of type (ω0 , θ0 ), if for each ω > ω0 , θ < θ0 , there is a constant M ≥ 1 such that |S(z)|B(X) ≤ M eωRez ,

z ∈ Σ(0, θ).

(7.2)

(ii) A resolvent S(t) for (7.1) is called analytic if the function S(·) : ⺢+ → B(X) ∩ B(Y ) admits analytic extension to a sector Σ(0, θ0 ), θ0 ≤ π/2. Then S(t) will be said to be of type (ω0 , θ0 ), if for each ω > ω0 , θ < θ0 there is a constant M ≥ 0 such that |S(z)|B(X) + |S(z)|B(Y ) ≤ M eωRez ,

z ∈ Σ(0, θ).

(7.3)

Obviously, Corollary 2.1 remains valid for weak analytic resolvents of type (ω0 , θ0 ) since its proof only involves Cauchy’s integral formula and Estimate (7.2). If S(t) is an analytic resolvent we have even |S (n) (t)|B(X) + |S (n) (t)|B(Y ) ≤ M n!eωt(1+α) (αt)−n ,

t > 0, n ∈ ⺞

(7.4)

where M = M (ω, θ), α = sin θ, and ω > ω0 , θ < θ0 . Theorem 2.1 can be extended directly to the case of nonscalar equations of the form (7.1). ∞ Theorem 7.1 Suppose A ∈ L1loc (⺢+ ; B(Y, X)) and 0 |A(t)|B(Y,X) e−ωA t dt < ∞, for some ωA ≥ 0. Then Equation (7.1) admits an analytic resolvent S(t) of type (ω0 , θ0 ) if and only if the following conditions are satisfied. J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_7, © Springer Basel 1993

185

186


ˆ ˆ (AN1) There is λ0 > ωA such that I − A(λ) is injective, the part of I − A(λ) in −1 ˆ Y is surjective, and (I − A(λ)) is bounded in X and in Y, for each λ > λ0 ; we −1 /λ, λ > λ . ˆ let H(λ) = (I − A(λ)) 0 (AN2) The family {H(λ)}λ>λ0 ⊂ B(X) ∩ B(Y ) admits analytic extension to the sector Σ(ω0 , π/2 + θ0 ). (AN3) For each ω > ω0 , θ > θ0 there is a constant C = C(ω, θ) such that |H(λ)|B(X) + |H(λ)|B(Y ) ≤ C/|λ − ω|,

λ ∈ Σ(ω, π/2 + θ).

(7.5)

Proof: Necessity. Let S(t) be an analytic resolvent of type (ω0 , θ) of (7.1). Theoˆ rem 6.2 implies then (AN1) as well as H(λ) = S(λ) for λ > λ0 . As in Section 2.1, Cauchy’s formula and the holomorphy of S(t) in X and in Y yield ∞ iθ ˆ S(λ) = S(teiθ )e−λte eiθ dt, λ > ω0 , |θ| < θ0 . 0

This gives the analytic continuation of H(λ) to Σ(ω0 , π/2 + θ0 ) as well as the estimate (7.5), i.e (AN2) and (AN3) hold. Sufficiency. Conversely, suppose (AN1) ∼ (AN3) hold. Then (N1) of Theorem 6.2 holds for λ > λ0 , and Cauchy’s formula yields the representation (−1)n n! (n) H (λ) = H(z)(z − λ)−(n+1) dz, λ > ω, n ∈ ⺞, (7.6) 2πi Γ where ω > ω0 , θ < θ0 , and Γ denotes the contour {ω + (−∞, R]e−i(π/2+θ) } ∪ {ω + Rei[−π/2−θ,π/2+θ] } ∪ {ω + [R, ∞)ei(π/2+θ) }. Choosing e.g. R = (λ − ω)/(n + 1) a direct estimate then yields (N2), hence there is a pseudo-resolvent S(t) for (7.1), by Theorem 6.2. However, by uniqueness of the Laplace transform, as in the proof of Theorem 2.1, S(t) is given by the contour integral 1 S(z) = eλz H(λ)dλ, | arg z| < θ0 , (7.7) 2πi Γ and this representation formula yields analyticity of S(z) in Σ(0, θ0 ), as well as the type estimates, in X and also in Y . 2

Similarly, one can also characterize analyticity of weak resolvents of (7.1). We state this as ∞ Corollary 7.1 Suppose A ∈ L1loc (⺢+ ; B(Y, X)) and 0 |A(t)|B(Y,X) e−ωA t dt < ∞. Then (7.1) admits a weak analytic resolvent of type (ω0 , θ0 ) if and only if the following conditions are satisfied. ˆ (WA1) There is λ0 > ωA such that I − A(λ) is injective and admits a left-inverse λH(λ) ∈ B(X) for each λ > λ0 .

7. Nonscalar Parabolic Equations

187

(WA2) The operator family {H(λ)}λ>λ0 ⊂ B(X) admits analytic extension to Σ(ω0 , π/2 + θ0 ). (WA3) For each ω > ω0 , θ > θ0 there is a constant C = C(ω, θ) such that |H(λ)|B(X) ≤ C/|λ − ω|,

λ ∈ Σ(ω, π/2 + θ).

(7.8)

The proof is similar to that of Theorem 7.1 based on (7.6) and (7.7) in X and is therefore omitted. In practice, the analytic extensions of H(λ) required in (AN2) and (WA2) are ˆ obtained by extending A(λ) analytically to the required sector, and then to study ˆ invertibility of I − A(λ) in X and in Y . As seen in Corollary 2.3, this usually will only work if the family {A(t)}t>0 ⊂ B(Y, X) itself has some very strong smoothness properties, like analyticity in Σ(0, θ0 ). Next we consider a-regularity of weak analytic resolvents. Corollary 7.2 Under the assumptions of Corollary 7.1, let in addition a ∈ L1loc (⺢+ ) be of exponential growth, such that a ˆ(λ) admits analytic extension to Σ(ω0 , π/2 + θ0 ), and that in addition the following condition holds. (AN4) For each ω > ω0 , θ > θ0 there is a constant C = C(ω, θ) such that |ˆ a(λ)H(λ)|B(X,Y ) ≤ C/|λ − ω|,

λ ∈ Σ(ω, π/2 + θ).

(7.9)

Then the weak analytic resolvent S(t) for (7.1) is a-regular. T (t) = (a ∗ S)(t) admits analytic extension to Σ(0, θ0 ), and for each ω > ω0 , θ > θ0 there is a constant M > 0 such that |T (z)|B(X,Y ) ≤ M eωRez ,

z ∈ Σ(0, θ).

(7.10)

The proof employs the same arguments as that of Theorem 7.1 and is therefore omitted. If (7.1) admits an analytic resolvent, the Cauchy estimates (7.4) show that the arguments in the proof of Theorem 2.4 still can be applied. This leads to the following result on maximal regularity of type C α for (7.1). Corollary 7.3 Let S(t) be a weak analytic resolvent for (7.1), f : J → X continuous, J = [0, T ], and let u(t) formally be defined by the variation of parameters formula t ˙ − s)f (s)ds, t ∈ J. u(t) = f (t) + S(t (7.11) 0

then u ∈ C0α (J; X); if in addition S(t) is a pseudoThen (i) If f ∈ resolvent then u is a mild solution of (7.1). (ii) If f ∈ C0α (J; Y ) and if S(t) is an analytic resolvent for (7.1) then u ∈ C0α (J; Y ) and u is a strong solution of (7.1). C0α (J; X)

188


(iii) If (AN4) holds and if f = a ∗ g with g ∈ C0α (J; X) then u ∈ C0α (J; X) is a strong solution of (7.1). For the proof one only has to follow the proof of Theorem 2.4 to obtain u ∈ C α in X resp. Y, and then apply the usual approximation arguments. 7.2 Parabolic Equations For parabolic equations of the form (7.1) the methods leading to Theorem 3.1 can also be employed, we briefly discuss this now. Definition 3.1 is extended to nonscalar equations by the following Definition 7.2 Let A ∈ L1loc (⺢+ ; B(Y, X)) be of subexponential growth. (7.1) is called weakly parabolic if the following two conditions are satisfied ˆ (NP1) For each Re λ > 0, the operator A(λ) with domain Y is closed in X, and ˆ I − A(λ) is invertible in X. (NP2) There is a constant M ≥ 1 such that −1 ˆ |B(X) ≤ M, |(I − A(λ))

for all Re λ > 0.

(7.1) is called parabolic if it is weakly parabolic and in addition (NP3) There is a constant M ≥ 1 such that −1 ˆ |B(Y ) ≤ M, |(I − A(λ))

for all Re λ > 0,

is valid. The second notion needed for the announced extension of Theorem 3.1 is introduced in Definition 7.3 Let A ∈ L1loc (⺢+ ; B(Y, X)) be of subexponential growth and supˆ pose A(λ) is closed with domain Y, for each Re λ > 0; let k ∈ ⺞. Then A(t) is called weakly k-regular, if there is a constant C > 0 such that ˆ |λn Aˆ(n) (λ)y|X ≤ C(|y|X + |A(λ)y| X ),

Re λ > 0, y ∈ Y, n ≤ k.

(7.12)

holds. A(t) is called k-regular if it is weakly k-regular and ˆ |λn Aˆ(n) (λ)|Y y|Y ≤ C(|y|Y + |A(λ)| Y y|Y ),

ˆ Re λ > 0, y ∈ D(A(λ)| Y ), n ≤ k. (7.13)

is satisfied for some constant C ≥ 0. Here Aˆ(n) (λ)|Y denotes the part of Aˆ(n) (λ) in Y, as before. Obviously, in the scalar case A(t) = a(t)A, Y = XA , A(t) is k-regular if a(t) is k-regular in the sense of Definition 3.3. Theorem 7.2 Let A ∈ L1loc (⺢+ ; B(Y, X)) be of subexponential growth, and assume that (7.1) is parabolic and k-regular for some k ≥ 1. Then there is a resolvent


189

S ∈ C k−1 ((0, ∞); B(X) ∩ B(Y )) for (7.1), and there is a constant N ≥ 1 such that |tn S (n) (t)|B(X)∩B(Y ) ≤ N, t ≥ 0, n ≤ k − 1, (7.14) and |tk S (k−1) (t) − sk S (k−1) (s)|B(X)∩B(Y ) ≤ N |t − s|[1 + log

t ], t−s

0 ≤ s < t. (7.15)

If (7.1) is merely weakly parabolic and A(t) weakly k-regular for some k ≥ 1, then there is a weak resolvent S ∈ C k−1 ((0, ∞); B(X)) for (7.1), and (7.14), (7.15) hold in B(X), instead of B(X) ∩ B(Y ). Proof: The proof of Theorem 7.2 is based on Theorem 6.2 and results on the inversion of the Laplace transform in Section 0. As before we let −1 ˆ H(λ) = (I − A(λ)) /λ,

Re λ > 0,

which exists by weak parabolicity and satisfies the parabolic estimate |H(λ)|B(X) ≤

M , |λ|

Re λ > 0.

For H (λ) we have −1 ˆ H (λ) = −H(λ)/λ + H(λ)Aˆ (λ)(I − A(λ)) ,

Re λ > 0,

hence by weak 1-regularity and weak parabolicity |H (λ)|B(X) ≤

M , |λ|2

Re λ > 0.

By induction we obtain |H (n) (λ)|B(X) ≤

Mn , |λ|n+1

Re λ > 0, n ≤ k,

for some constants Mn . Theorem 0.4 now implies |H (n) (λ)|B(X) ≤

M0 n! , λn+1

for all λ > 0, n ∈ ⺞0 ,

and as in the proof of Theorem 6.4 this yields a weak resolvent S(t) for (7.1), which by Theorem 0.4 satisfies estimates (7.14), (7.15) in B(X). If (7.1) is even parabolic and k-regular, we may repeat the arguments just given in Y, and conclude the existence of a pseudo-resolvent S(t) by Theorem 6.2 which is subject to (7.14), (7.15), and a resolvent by Theorem 0.4. 2

By similar arguments also a-regularity of (weak) resolvents for parabolic problems can be obtained. Without proof we state this as

190


Corollary 7.4 Assume that (7.1) is weakly parabolic, A(t) weakly k-regular for some k ≥ 1, and let a ∈ L1loc (⺢+ ) be k-regular and such that −1 ˆ (NP4) |ˆ a(λ)(I − A(λ)) |B(X,Y ) ≤ M, Re λ > 0, for some constant M . Then the weak resolvent S(t) for (7.1) is a-regular and T (t) = (a ∗ S)(t) satisfies in addition |tn T (n) (t)|B(X,Y ) ≤ N, t ≥ 0, n ≤ k − 1, (7.16) and |tk T (k−1) (t) − sk T (k−1) (s)|B(X,Y ) ≤ N |t − s|[1 + log

t ], t−s

0 ≤ s < t, (7.17)

with some constant N ≥ 1. The (weak) resolvent constructed in Theorem 7.2 can be used as in the proof of Theorem 3.3 to obtain maximal regularity of type C α . Corollary 7.5 Let the assumptions of Theorem 7.2 be satisfied with k = 2, f : J → X be continuous, and let u(t) formally be defined by (7.11). Then (i) If f ∈ C0α (J; X) then u ∈ C α (J; X) is a mild solution of (7.1). (ii) If f ∈ C0α (J; Y ) then u ∈ C α (J; Y ) is a strong solution of (7.1). (iii) If (7.1) is weakly parabolic and A(t) weakly 2-regular, then f ∈ C0α (J; X) implies u ∈ C α (J; X). (iv) If the assumptions of Corollary 7.4 hold, then f = a ∗ g, g ∈ C0α (J; X) implies that u ∈ C α (J; Y ) is a strong solution of (7.1). 7.3 Parabolic Problems of Variational Type d

As in Section 6.7, let V and H denote Hilbert spaces such that V → H, and let ((·, ·)) resp. (·, ·) denote the inner products in V resp. H, and || · || resp. | · | the ∗ corresponding norms. Again we identify the antidual H of H with H and by ∗

d

duality we also get H → V ; the relation between the duality < ·, · > of V, V and the inner product in H is (v, h) = < v, h >,

∗

for all v ∈ V, h ∈ H.

∗

As before, the norm in V will be denoted by || · ||∗ . Let α : ⺢+ × V × V → ⺓ be such that α(t, ·, ·) is a bounded sesquilinear form on V for a.a. t ≥ 0. α is said to be locally integrable, if there is α0 ∈ L1loc (⺢+ ) such that |α(t, u, v)| ≤ α0 (t)||u|| ||v|| for all u, v ∈ V, t ∈ E0 , (7.18) where E0 has Lebesgue-measure zero. α is called uniformly measurable if for each T > 0, ε > 0 there are N + 1 measurable disjoint subsets Ej of J = [0, T ] with ∪N j=0 Ej = J, mes(E0 ) < ε, and bounded sesquilinear forms αj on V such that |α(t, u, v) −

N j=1

χj (t)αj (u, v)| ≤ ε||u|| ||v||

for all u, v ∈ V, t ∈ J \ E0 . (7.19)


191 ∗

By the Riesz-representation theorem there are operators A(t) ∈ B(V, V ) such that α(t, u, v) = − < u, A(t)v > for all u, v ∈ V, a.a.t ≥ 0; (7.20) It is readily verified that uniform measurability and local integrability of α imply ∗ A ∈ L1loc (⺢+ ; B(V ; V )). Consider the Volterra equation of variational type (6.54), i.e.

t

(w, u(t)) + 0

α(t − s, w, u(s))ds = < w, f (t) >,

∗

t ∈ J, w ∈ V,

(7.21)

∗

where f ∈ C(J; V ). As in Section 6.7, with X = V , Y = V, and A(t) defined by (7.20), Equation (7.21) can be rewritten in the form (7.1). Assuming a parabolic coercive estimate as well as some regularity of α, the results of Sections 7.1 and 7.2 can be directly applied to (7.21). The main assumptions will be stated in the frequency domain; they read (PV1) There is a function ϕ : Σ(0, θ) → (0, ∞) such that |ˆ α(λ, u, v)| ≤ ϕ(λ)||u|| ||v||,

for all u, v ∈ V, λ ∈ Σ(0, θ).

(PV2) There is a constant γ > 0 such that |ˆ α(λ, u, u)| ≥ γϕ(λ)||u||2 ,

for all u ∈ V, λ ∈ Σ(0, θ).

(PV3) There is φ ∈ (0, π) such that | arg α(λ, ˆ u, u)| ≤ φ,

for all u ∈ V, λ ∈ Σ(0, θ).

Here we tacitly assumed that α0 ∈ L1loc (⺢+ ) appearing in (7.18) is of supexponential growth, so that α(λ, ˆ u, v) is well-defined for Re λ > 0, at least. We want to stress that the function ϕ(λ) appearing in (PV1) and (PV2) is meant to be the same; this will be crucial in the sequel. Assume now that (PV1), (PV2), (PV3) are satisfied. We are going to show that (7.1) resulting from the variational equation (7.21) becomes parabolic in the sense of Definition 7.2. Since for each t ≥ 0 and z ∈ Σ(0, π) we have with ψ = arg z |t + z|2 = t2 + |z|2 + 2t|z| cos ψ ≥ c(ψ)2 (t2 + |z|2 ),

where c(ψ) =

1 √

2 cos(ψ/2)

for |ψ| ≤ π/2 for |ψ| > π/2,

(PV2) and (PV3) yield ˆ ||u|| · ||tu − A(λ)u|| ∗

¯ u, u)| ˆ ≥ | < u, tu − A(λ)u > | = | t|u|2 + α( ˆ λ, ≥ c(φ)(t2 |u|2 + |ˆ α(λ, u, u)|2 )1/2 2 ¯ ≥ c(φ)γϕ(λ)||u|| , for all u ∈ V, λ ∈ Σ(0, θ).

192


ˆ Therefore, for t ≥ 0 the operators t − A(λ) are bijective and boundedly invertible, we obtain the estimates −1 ˆ |(t − A(λ)) |B(V ∗ ,V ) ≤

1 , ¯ γϕ(λ)c(φ)

λ ∈ Σ(0, θ), t ≥ 0.

(7.22)

On the other hand, (PV1) implies ˆ ∗ |A(λ)| B(V,V ) =

sup ||u||,||v||≤1

ˆ | < u, A(λ)v >|=

sup ||u||,||v||≤1

¯ u, v)| ≤ ϕ(λ), ¯ (7.23) |ˆ α(λ,

hence by (7.22), 1 + γc(φ) −1 ˆ |(t − A(λ)) |B(V ∗ ) ≤ γc(φ)t

λ ∈ Σ(0, θ), t ≥ 0.

(7.24)

Combining (7.24) with (7.23), and (7.22) for t = 0 yields −1 −1 −1 ˆ ˆ ˆ ˆ |(t − A(λ)) |B(V ) = |(A(λ)) (t − A(λ)) A(λ)|B(V ) −1

ˆ ≤ |A(λ)

−1 ˆ ˆ ∗ |B(V ∗ ,V ) · |(t − A(λ)) |B(V ∗ ) · |A(λ)| B(V,V )

−1 ¯ ¯ = 1 + γc(φ) , ≤ (γϕ(λ)c(φ)) (1 + γc(φ))(γc(φ)t)−1 ϕ(λ) t(γc(φ))2

t > 0,

i.e. −1 ˆ |(t − A(λ)) |B(V ) ≤ c1 (φ)/t,

λ ∈ Σ(0, θ), t > 0.

(7.25)

In particular, (7.24) and (7.25) with t = 1 imply the parabolic estimates, provided θ ≥ π/2; if θ > π/2 we even see that the assumptions of Theorem 7.1, in particular (7.5), with ω0 = 0, θ0 = θ − π/2 are satisfied, hence in this situation we obtain an analytic resolvent of type (0, θ0 ) for (7.21), without any further assumption. An interesting special case arises if ϕ(λ) can be chosen as ¯ = |ˆ ϕ(λ) a(λ)|,

λ ∈ Σ(0, θ),

(7.26)

for some a ∈ L1loc (⺢+ ) of subexponential growth. Then (7.22) implies |ˆ a(λ)H(λ)|B(V ∗ ,V ) ≤ c2 (φ)/|λ|,

λ ∈ Σ(0, θ),

(7.27)

which is the basic estimate for a-regularity of the resolvent for (7.21). In case θ = π/2 only, Theorem 7.2 can be applied if in addition A(t) is kregular for some k ≥ 1. For the variational setting this assumption seems to be too stringent; however, the condition (PV4) there is C > 0 such that |λn (

d n ) α ˆ (λ, u, v)| ≤ Cϕ(λ)||u|| ||v||, for all u, v ∈ V, Re λ > 0, 1 ≤ n ≤ k, dλ


193

is easily seen to imply weak k-regularity of A(t), by (PV2). It turns out that (PV4) is in fact enough to obtain a resolvent, as a small modification of the proof of Theorem 7.2 shows; namely to estimate |H(λ)|B(Y ) we write −1 ˆ H (λ) = −H(λ)/λ + H(λ)Aˆ (λ)(I − A(λ))

and obtain with the aid of (7.25), (PV4), and (7.22) |H (λ)|B(Y )

≤ |H(λ)|B(Y ) /|λ| + |H(λ)|B(V ∗ ,V ) |Aˆ (λ)|B(V,V ∗ ) |λH(λ)|B(V ) −1 ¯ ¯ ≤ |c1 (φ)|/|λ|2 + (γϕ(λ)c(φ)) (Cϕ(λ)/|λ|)c 1 (φ)/|λ|

= c3 (φ)/|λ|2 . Similarly, the estimates for H (n) (λ), n ≤ k, are obtained, and then the remaining part of the proof of Theorem 7.2 applies again. Summarizing we have the following result on resolvents and maximal regularity of type C α for parabolic problems of variational type. Theorem 7.3 Let α : ⺢+ × V × V → ⺓ be such that α(t, ·, ·) is a bounded sesquilinear form on V for a.a. t ≥ 0, uniformly measurable and locally integrable with α0 from (7.18) of subexponential growth. (i) Suppose α(λ, ˆ u, v) admits analytic extension to the sector Σ(0, θ), θ > π/2, and assume (PV1), (PV2) and (PV3) on Σ(0, θ). Then there is an analytic resol∗ vent S(t) for (7.21) in V of type (0, θ0 ), where θ0 = θ − π/2. (ii) Assume (PV1), (PV2) and (PV3) with θ = π/2, as well as (PV4) for some ∗ k ≥ 1. Then there is a resolvent S(t) for (7.21) in V of class C (k−1) on (0, ∞), ∗ and Estimates (7.14) and (7.15) hold in V and V . (iii) If ϕ(λ) = |ˆ a(λ)|, λ ∈ Σ(0, θ), for some k-regular a ∈ L1loc (⺢+ ) then the ∗ resolvent S(t) for (7.21) in V is a-regular, and (a ∗ S)(t) is of class C k−1 on ∗ (0, ∞) in B(V , V ). (iv) In case (i) and in case (ii) for k ≥ 2, (7.21) has the maximal regularity property of type C α , α ∈ (0, 1), as stated e.g. in Corollary 7.5. 7.4 Maximal Regularity of Perturbed Parabolic Problems In the remainder of Section 7 we consider the perturbed problem t t s u(t) = A(t−τ )u(τ )dτ + a(t−s)( dB(τ )u(s−τ ))ds+f (t), 0

0

0

t ∈ J, (7.28)

where a ∈ L1loc (⺢+ ), A ∈ L1loc (⺢+ ; B(Y, X)), B ∈ BV (⺢+ ; B(Y, X)) is such that B(0) =B(0+) = 0, f ∈ C(J; X), and J = [0, t0 ]. The unperturbed problem (7.1) is assumed to admit an a-regular resolvent S0 ∈ C 1 ((0, ∞); B(X) ∩ B(Y )) such that |S0 (t)|B(X)∩B(Y ) + |tS˙ 0 (t)|B(X)∩B(Y ) ≤ N,

t ∈ J,

(7.29)

|t − s| t [1 + log ], 0 ≤ s < t ≤ t0 , |S˙ 0 (t) − S˙ 0 (s)|B(X)∩B(Y )) ≤ N ts t−s

194


and T0 = a ∗ S0 ∈ C 1 ((0, ∞); B(X, Y )) is subject to |T0 (t)|B(X,Y ) + |tT˙0 (t)|B(X,Y ) ≤ N,

t ∈ J,

(7.30)

|t − s| t [1 + log ], 0 ≤ s < t ≤ t0 . ts t−s The resolvents obtained in Theorems 2.1 and 7.1, as well as those from Theorems 3.1 and 7.2 for k ≥ 2 have these properties and are good candidates for the sequel. An important special case is A(t) ≡ A the generator of an analytic semigroup, and a(t) ≡ 1; then S0 (t) = eAt , Y = XA , and (7.28) is equivalent to t u(t) ˙ = Au(t) + dB(τ )u(t − τ ) + g(t), u(0) = u0 , t ∈ J, |T˙0 (t) − T˙0 (s)|B(X,Y ) ≤ N

0

where g(t) = f˙(t) and u0 = f (0). We begin with an extension of the maximal regularity results of type C α to the perturbed problem (7.28). Theorem 7.4 Under the assumptions stated above, (7.28) has the maximal regularity property of type C α . More precisely, with α ∈ (0, 1), (i) if f ∈ C0α (J; X) there is a unique mild solution u ∈ C0α (J; X) of (7.28); (ii) if f ∈ C0α (J; Y ) there is a unique strong solution u ∈ C0α (J; Y ) of (7.28); (iii) if f = a ∗ g, g ∈ C0α (J; X) there is a unique strong solution u ∈ C0α (J; Y ) of (7.28). Proof: By the variation of parameters formula, (7.28) is equivalent to d d (S0 ∗ f )(t) + (S0 ∗ a ∗ dB ∗ u)(t), dt dt The latter will be considered as a linear equation u(t) =

t ∈ J.

u = u0 + Ku in the Banach space U =

C0α (J; Y

||u||Y,α,ω =

(7.31)

(7.32)

) with norm

sup 0≤s0

1 ≤ p < ∞, 1 ≤ q ≤ ∞, α ∈ (0, 1). It is well known that Bpα,q (J; X) = (Lp (J; X), W01,p (J; X))α,q , are the real interpolation spaces between Lp (J; X) and W01,p (J; X); cp. Butzer and Berens [36]. Similarly, we let α,q B∞ (J; X) = (C0 (J; X), C01 (J; X))α,q ,

α ∈ (0, 1), 1 ≤ q ≤ ∞,

(7.41)

and with this notation we have α,∞ B∞ (J; X) = C0α (J; X),

α ∈ (0, 1).

Observe that the index 0 always refers to vanishing at t = 0. The following maximal regularity result is a considerable extension of Theorem 7.4. Theorem 7.5 Under the assumptions stated above, let S : f → u denote the solution map of (7.28), and let α ∈ (0, 1), p, q ∈ [1, ∞]. Then S enjoys the following properties. (i) S ∈ B(Bpα,q (J; X)) ∩ B(Bpα,q (J; Y )); (ii) a ∗ S = S(a ∗ ·) ∈ B(Bpα,q (J; X), Bpα,q (J; Y )). Proof: The proof parallels that of Theorem 7.4, the estimates (7.34) on dB ∗ u and (7.35), (7.36) being the main ingredients. Therefore we only have to establish the analogues of these estimates in Bpα,q (J; X). Choosing the norm on the base spaces Lp (J; X) resp. C0 (J; X) as t0 |f |p,ω = ( |f (t)|p e−ωpt dt)1/p , 1 ≤ p < ∞, 0

resp.

|f |∞,ω = sup |f (t)|e−ωt , 0 1. Since ∞ t0 ∞ (1−α)q−1 (h−α ϕ(h))q dh/h ≤ 2N t−1 h dh + 2N h−αq−1 dh < ∞, 0 0

and

0

∞

∞

sα ψ(s)ds/s = N

0

0

t0

(1 + s)−1 (1 + log(1 + s))sα−1 ds < ∞,

there follows the estimate X [u]X α,q;p ≤ c(α, t0 , q)[f ]α,q;p ,

f ∈ Bpα,q (J; X),

with some constant c(α, t0 , q) < ∞, thereby proving the theorem.

(7.43) 2

7.5 Resolvents for Perturbed Parabolic Problems Consider again (7.28) under the assumptions stated at the beginning of Section

198


7.4. We now show that there is an a-regular resolvent for (7.28). The resolvent will be constructed as follows. If (7.28) admits a resolvent S(t) then it must be the solution of d (a ∗ S0 ∗ dB ∗ S)(t), t ∈ ⺢+ . (7.44) dt From (7.44) we obtain with T = a ∗ S the following three integral equations. S(t) = S0 (t) +

T (t) = T0 (t) + (T˙0 ∗ dB ∗ T )(t),

t ∈ ⺢+ .

d (S0 ∗ dB ∗ T )(t), t ∈ ⺢+ . dt S(t) = S0 (t) + (T˙0 ∗ dB ∗ S)(t), t ∈ ⺢+ .

S(t) = S0 (t) +

(7.45) (7.46) (7.47)

1

Since T0 ∈ C ((0, ∞); B(X, Y )), (7.45) will turn out to admit a solution with values in B(X, Y ). (7.46) is merely the definition of S(t) in B(X) after T (t) is known to exist, and (7.47) shows that S(t) is leaving Y invariant. To solve (7.45) in B(X, Y ) we employ the maximal regularity result of type Bpα,q , Theorem 7.4. This approach works, since by (7.30) |T0 (t) − T0 (t − h)|B(X,Y ) ≤ N min{2,

h }, t−h

t > h,

hence |T0 (·) − T0 (· − h)|p

≤ N[ = N [2

3 2h

0

p−1

2p dt + hp

∞ 3 2h

dt ]1/p (t − h)p

· 3h + hp · (h/2)1−p (p − 1)−1 ]1/p = Cp h1/p ,

1/p,∞

i.e. T0 (·) ∈ Bp (J; B(X, Y )) for each p ∈ (1, ∞). Similarly, one can show T0 ∈ Bpα,q (J; B(X, Y )) whenever αp < 1, p, q ∈ [1, ∞], α ∈ (0, 1). Clearly this excludes p = ∞, and so by means of this method one cannot obtain uniform L∞ bounds on T (t), for this a different approach must be used. The L∞ -bounds are quite delicate to obtain, Section 7.6 is devoted to this aim. Here we construct the resolvent and show its continuity properties assuming L∞ -bounds. The result reads as follows. Theorem 7.6 Let A ∈ L1loc (⺢+ ; B(Y, X)), B ∈ BVloc (⺢+ ; B(Y, X)) satisfy B(0) = B(0+) = 0, b(t) =VarB|t0 , and let a ∈ L1loc (⺢+ ). Assume that (7.1) admits an a-regular resolvent S0 ∈ C 1 ((0, ∞); B(X) ∩ B(Y )) with T0 = a∗ S0 ∈ C 1 ((0, ∞); B(X, Y )) such that (7.29) and (7.30) hold on each interval J = [0, t0 ], with constant N = N (t0 ). Then there is an a-regular resolvent S(t) for (7.28). S(t) and T (t) = (a ∗ S)(t) have the following additional properties. ∞ (i) S ∈ L∞ loc (⺢+ ; B(X) ∩ B(Y )), T ∈ Lloc (⺢+ ; B(X, Y )); (ii) S(t) resp. T (t) is left-continuous in B(X) ∩ B(Y ) resp. in B(X, Y ) on (0, ∞); (iii) S(t) resp. T (t) is right-continuous in B(X)∩B(Y ) resp. in B(X, Y ) for t ∈ Nr , where Nr denotes the set of jump points of the resolvent kernel r of b;


199

(iv) S and T have the following additional regularity:

S∈

1/p,∞

Bp,loc (⺢+ ; B(X) ∩ B(Y )),

p∈(1,∞)

and

T ∈

1/p,∞

Bp,loc (⺢+ ; B(X, Y )).

p∈(1,∞)

Note that the resolvent kernel r(t) of b(t) is defined as the solution of r = b +db ∗ r; therefore Nr = {t > 0 : r(t) = r(t+)} is the additive semimodule generated by the jump points of b, i.e. Nb = {t > 0 : b(t) = b(t+)}. Recall our convention that BV -functions b are always normalized by left-continuity and by b(0) = 0. Example 1.1 shows that Theorem 7.6 is optimal; the discontinuity of B(t) = Ae0 (t − 1) is reproduced in the resolvent S(t) of this example at tn = n, i.e. at each t ∈ Nr = ⺞. Proof of Theorem 7.6: (a) By the estimates derived in the proof of Theorem 7.5, i.e. (7.42) and (7.43) and its analogs in Y and for T˙0 ∗ f , it is readily seen that 1/p,∞ there is a unique solution T ∈ p∈(1,∞) Bp,loc (⺢+ ; B(X, Y )) of (7.45) as well as 1/p,∞ a unique solution S1 ∈ p∈(1,∞) Bp,loc (⺢+ ; B(Y )) of (7.47). Then S defined by 1/p,∞ (7.46) belongs to p∈(1,∞) Bp,loc (⺢+ ; B(X)). Integrating (7.45), (7.46) and (7.47) and convolving the latter two with a(t) we obtain 1 ∗ T = 1 ∗ T0 + T0 ∗ dB ∗ T

in B(X, Y ),

1 ∗ a ∗ S = 1 ∗ a ∗ S0 + a ∗ S0 ∗ dB ∗ T 1 ∗ a ∗ S1 = 1 ∗ a ∗ S0 + a ∗ T0 ∗ dB ∗ S1

in B(X), in B(Y ),

d

hence by uniqueness and by Y → X we obtain 1 ∗ T y = 1 ∗ a ∗ Sy = 1 ∗ a ∗ S1 y for each y ∈ Y, i.e. S1 = S|Y , and T = a ∗ S; this proves (iv). It is also clear from the construction that S satisfies the resolvent equations for (7.28), e.g. in integrated form. ∞ (b) Let ρε ∈ C0∞ (⺢) be nonnegative, supp ρε ⊂ (0, ε), −∞ ρε (t)dt = 1, and let Bε (t) =

0

∞

dB(τ )ρε (t − τ ),

t ∈ ⺢;

(7.48)

t then Bε ∈ C ∞ (⺢; B(Y, X)), supp Bε ⊂ (0, ∞), and 0 |Bε (τ )|dt ≤ b(t), as well as t0 −ωt t e |Bε (t)|dt ≤ 0 0 db(τ )e−ωτ . 0 By Theorem 6.1, there is an a-regular resolvent Sε (t) of (7.28) with dB replaced by Bε , and as solution of (7.45)∼(7.47), with dB replaced by Bε , Sε and Tε = a∗Sε also have property (iv). In Section 7.6 we shall prove the following

200


L∞ -estimates: For each t0 > 0 there is a constant C = C(t0 ) such that |Sε (t)|B(X) + |Sε (t)|B(Y ) + |Tε (t)|B(X,Y ) ≤ C,

t ∈ J, ε ∈ (0, 1).

Assume that these L∞ -estimates are true. Then we have T − Tε = T˙0 ∗ (dB ∗ Tε − Bε ∗ Tε ) + T˙0 ∗ dB ∗ (T − Tε ) d S − Sε = (S0 ∗ (dB ∗ T − Bε ∗ Tε )) in B(X), dt S − Sε = T˙0 ∗ (dB ∗ Sε − Bε ∗ Sε ) + T˙0 ∗ dB ∗ (S − Sε ) Since

t 0

1/p,∞

|Bε (t)|B(Y,X) dτ ≤ b(t), t > 0, {Tε }ε∈(0,1) ⊂ Bp

in B(X, Y ), (7.49) in B(Y ).

(J; B(X, Y )) is uni-

1/p,∞ formly bounded, {dB ∗ Tε }ε∈(0,1) ⊂ Bp (J; B(X)) is as well, and therefore dB ∗ Tε − ρε ∗ dB ∗ Tε → 0 as ε → 0 in each Bpα,∞ (J; B(X, Y )), αp < 1. By (7.49) this implies Tε → T in Bpα,∞ (J; B(X, Y )), for each αp < 1, in particular Tε → T in Lp (J; B(X, Y )). Similarly we also obtain Sε → S in Lp (J; B(X) ∩ B(Y )), where

J = [0, t0 ], t0 > 0 being arbitrary. Passing to a subsequence εn → 0 if necessary, we obtain a.e. convergence, thereby proving the L∞ -estimates also for S(t), i.e. (i) of Theorem 7.6 follows. (c) To prove that S(t) is an a-regular resolvent of (7.28) in the sense of Definition 6.2, observe that the first resolvent equation implies T ∈ C(⺢+ ; B(X)) and S ∈ C(⺢+ ; B(Y, X)); in particular, S(t)y can be extended as a continuous function in X for each y ∈ Y . Since Y is dense in X, the family {S(t)}t≥0 ⊂ B(X) is everywhere defined, and |S(t)|B(X) ≤ C(t0 ), for all t ≤ t0 , and so {S(t)}t≥0 is strongly continuous in X by the Banach-Steinhaus theorem. Similarly, the second resolvent equation shows T ∈ C(⺢+ ; B(Y )), and so by the same argument {T (t)}t≥0 is strongly continuous in B(X, Y ). This shows that S(t) is a resolvent for (7.28) in the sense of Definition 6.2. The proofs of the remaining continuity properties (ii) and (iii) do not follow from the L∞ -estimates alone and will be postponed until Section 7.6. 2

Let us pause to state some additional regularity properties of S(t) if B(t) has better properties, these follow directly from the integral equations (7.45), (7.46), (7.47), and Theorem 7.6. Corollary 7.6 Let the assumptions of Theorem 7.6 be satisfied. Then the resolvent S(t) of (7.28) has the following additional properties. (i) If B ∈ C(⺢+ ; B(Y, X)) then S − S0 ∈ C(⺢+ ; B(X) ∩ B(Y )) and T = a ∗ S ∈ C(⺢+ ; B(X, Y )). α (ii) If B ∈ C0α (⺢+ ; B(Y, X)) then S − S0 ∈ Cloc (⺢+ ; B(X) ∩ B(Y )) and T − T0 ∈ α Cloc (⺢+ ; B(X, Y )). As a consequence of the continuity properties of the resolvent S(t) for (7.28), observe that in Proposition 6.3 for (7.28) the conditions ‘f ∈ W 1,1 ’ in (ii) and (iii)


201

resp. ‘g ∈ W 1,1 ’ in (iv) can be replaced by ‘f ∈ BV ∩ C’ resp. ‘g ∈ BV ∩ C’. This follows from the fact that S ∗ df resp. S ∗ dg is now well-defined and continuous, as a short reflection shows. 7.6 Uniform Bounds for the Resolvent Throughout this subsection we use the same notation as in Sections 7.4 and 7.5. Our aim is to prove the L∞ -estimates for the resolvents Sε (t) of the approximating equations (7.28) with dB replaced by Bε , as stated in Section 7.5. This will be done by a term by term estimate of the Neumann-series corresponding to the solutions of (7.45), (7.46), and (7.47), i.e. Tε =

∞

(T˙0 ∗ Bε )∗n ∗ T0

in B(X, Y ),

(7.50)

n=0

Sε = S0 +

∞

(S˙ 0 ∗ Bε + Bε ) ∗ (T˙0 ∗ Bε )∗n ∗ T0

in B(X),

(7.51)

n=0

Sε =

∞

(T˙0 ∗ Bε )∗n ∗ S0

in B(Y ).

(7.52)

n=0

Since the arguments are the same for each of these series we shall concentrate only on the first one, (7.50). Claim 1 Let

Rn (t) = T˙0 ∗ Bε )∗n ∗ T0 (t)

for t ∈ J, n ∈ ⺞,

(7.53)

th

summand of the series (7.50) for Tε . Then t tn t2 t sn Rn (t) = ... 0 0 0 s2 τ τ T˙0 (t − sn )Bε (τn )T˙0 (sn − sn−1 )Bε (τn−1 ) (7.54) ...

denote the n

τ

. . . Bε (τ1 )T0 (s1 − τ )ds1 . . . dsn dτ1 . . . dτn n n holds, where tk = t − k τj and τ = 1 τj . Proof: Since the integrand is continuous, we may change the order of integration as we please. The induction step from n to n + 1 is verified as follows. t s ˙ ˙ T0 (t − s) Rn+1 (t) = T0 ∗ Bε ∗ Rn (t) = Bε (ρ)Rn (s − ρ)dρds 0 0 t t T˙0 (t − s)Bε (ρ)Rn (s − ρ)dsdρ = 0

ρ

t t

s−ρ

s−ρ−τn

=

... 0

ρ

0

0

202


s−ρ−

n 2

τj

0

...

s−ρ τ

sn

τ

s2

T˙0 (t − s)Bε (ρ)T˙0 (s − ρ − sn ).

τ

Bε (τ1 )T0 (s1 − τ )ds1 ds2 . . . dsn dτ1 . . . dτn dsdρ.

The transformation sj → sj + ρ and τ = τ + ρ yields t t s−ρ s−ρ−n τj s 2 Rn+1 (t) = ... 0

0

ρ

τ

0

sn τ

s2

τ

T˙0 (t − s)Bε (ρ)

. . . T˙0 (s − sn )Bε (τ1 )T0 (s1 − τ )ds1 . . . dsn dτ1 . . . dτn dsdρ, and commuting the integral over s with those over τ1 . . . τn leads to t Rn+1 (t) =

tn+1

t2

t

... 0

where tj = t − ρ −

s2

...

0

0

n

sn

τ

τ

τ

{. . .}ds1 . . . dsn dsdτ1 . . . dτn dρ

τk , i.e (7.54) holds for n + 1.

j

2

The representation (7.54) of Rn (t) suggests to consider t

sn

Zn (t) =

s2

... τ

τ

T˙0 (t − sn )Bε (τn ) . . . Bε (τ1 )T0 (s1 − τ )ds1 . . . dsn (7.55)

τ

first, where the dependence of Zn on τ1 , . . . , τn has been dropped. To obtain the desired estimates we decompose Zn (t) according to Claim 2 We have t t ˙ Zn (t) = T0 (t − s)Un (t, s)ds + T0 (t − s)Vn (t, s)ds + T0 (t − τ )Wn (t), (7.56) τ

τ

where Un , Vn , Wn are defined inductively by U1 (t, s) = Bε (τ1 )(T0 (s − τ ) − T0 (t − τ )), V1 (t, s) ≡ 0, W1 (t) = Bε (τ1 )T0 (t − τ ),

(7.57)

Wn+1 (t) = Bε (τn+1 )T0 (t − τ )Wn (t),

(7.58)

and Vn+1 (t, s) = Bε (τn+1 )[T˙0 (t − s)Un (t, s) + T0 (t − s)Vn (t, s)], Un+1 (t, s) = + +

Bε (τn+1 ){T0 (s − τ )Wn (s) − T0 (t − τ )Wn (t) s (T˙0 (s − r)Un (s, r) − T˙0 (t − r)Un (t, r))dr τ s [T0 (s − r)Vn (s, r) − T0 (t − r)Vn (t, r)]dr.} τ

(7.59)

(7.60)


203

Proof: For n = 1 we have

t

Z1 (t) =

T˙0 (t − s)Bε (τ1 )T0 (s − τ )ds

τ t

=

T˙0 (t − s)Bε (τ1 )(T0 (s − τ ) − T0 (t − τ ))ds

τ

t +( T˙0 (t − s)ds)Bε (τ1 )T0 (t − τ ), τ

hence (7.56), and (7.57) hold for n = 1. Let the assertions be true for n; then we have t Zn+1 (t) = T˙0 (t − s)Bε (τn+1 )Zn (s)ds τ t s ˙ = T0 (t − s)Bε (τn+1 ) T˙0 (s − r)Un (s, r)drds τ τ t s ˙ + T0 (t − s)Bε (τn+1 ) T0 (s − r)Vn (s, r)drds τ τ t + T˙0 (t − s)Bε (τn+1 )T0 (s − τ )Wn (s)ds τ t s ˙ = T0 (t − s)Bε (τn+1 ){ [T˙0 (s − r)Un (s, r) − T˙0 (t − r)Un (t, r)]dr τ τ s + [T0 (s − r)Vn (s, r) − T0 (t − r)Vn (t, r)]dr τ t +[T0 (s − τ )Wn (s) − T0 (t − τ )Wn (t)]}ds + T˙0 (t − s)Bε (τn+1 ) τ s ·{ [T˙0 (t − r)Un (t, r) + T0 (t − r)Vn (t, r)]dr + T0 (t − τ )Wn (t)}ds τ t = T˙0 (t − s)Un+1 (t, s)ds τ t + T0 (t − r)Vn+1 (t, r)dr + T0 (t − τ )Wn+1 (t), τ

hence the assertion also holds for n + 1. 2

To obtain the desired estimate for Zn (t) we have to estimate Un , Vn , Wn separately by induction. This is easy for Wn . Claim 3 Let αn =

n 1

|Bε (τj )|B(Y,X) . Then |Wn (t)| ≤ αn N n

for all t ≥ 0,

(7.61)

204


and |Wn (t) − Wn (s)| ≤ αn nN nϕ1 (

t−s ) s−τ

for all τ < s < t < t0 ,

(7.62)

where ϕ1 (a) = log(1 + a). Proof: We have by |W1 (t)| ≤ |Bε (τ1 )| · N ≤ α1 N and |W1 (t) − W1 (s)| ≤ |Bε (τ1 )||T0 (t − τ ) − T0 (s − τ )| ≤ α1 N log(1 +

t−s ), s−τ

i.e. (7.61) and (7.62) for n = 1. For the induction step we obtain with (7.58) |Wn+1 (t)| ≤ |Bε (τn+1 )|N |Wn (t)| ≤ αn+1 N n+1 , and |Wn+1 (t) − Wn+1 (s)|

≤ |Bε (τn+1 )||T0 (t − τ ) − T0 (s − τ )||Wn (t)| + |Bε (τn+1 )||T0 (s − τ )||Wn (t) − Wn (s)| t−s ≤ |Bε (τn+1 )|αn log(1 + )(N n+1 + N nN n) s−τ t−s = αn+1 (n + 1)N n+1 ϕ1 ( ). s−τ 2

To derive the estimates for Un and Vn we define functions ϕn (a) inductively by ϕn+1 (a) =

0

1

ϕn (

σ a σ dσ ) log(1 + )[1 + log(1 + )] . 1−σ σ a σ

(7.63)

Some properties of ϕn are collected as Claim 4 Given n ∈ ⺞, we have (i) 0 ≤ ϕn (a) ≤ c(n, β)aβ , a ≥ 0, β ∈ (0, 1); (ii) ϕn (a) ≤ ϕn+1 (a) for a ≥ 0; (iii) ϕn (a) is increasing for a ≥ 0; (iv) ϕn (a)/a is decreasing for a ≥ 0. Proof: By means of the elementary inequality a ≤ log(1 + a) ≤ aβ /β, 1+a

a > 0,

(7.64)


205

we obtain for the induction step 0

≤ ϕn+1 (a) 1 σ β+δ a β −1 a σ dσ ≤ c(n, β + δ)( ) [( ) β + ( )β+δ ( )δ (β + δ)−1 δ −1 ] 1 − σ σ σ a σ 0 1 = c(n, β + δ) = (β −1 + (β + δ)−1 δ −1 ) (1 − σ)−(β+δ) σ δ−1 dσ 0

≤ c(n + 1, β)aβ , where δ > 0 is such that β + δ < 1. Again using (7.64) we get ϕ2 (a)

1

=

log(1 + 0

σ a σ dσ ) log(1 + )[1 + log(1 + )] 1−σ σ a σ 1

σ/(1 − σ) dσ · σ 0 1 + σ/(1 − σ) = log(1 + a) = ϕ1 (a); ≥ log(1 + a)

hence (ii) follows inductively. (iii) and (iv) are implied by log(1+x)(1+log(1+1/x)) increasing and x−1 log(1 + x)[1 + log(1 + 1/x)] decreasing for x > 0. 2

By means of the function ϕn (a) we obtain the following estimates for Un and Vn . Claim 5 Let C ≥ 6N . Then for each n ∈ ⺞ we have |Un (t, s)| ≤ C n αn ϕn (

t−s ), s−τ

|Vn (t, s)| ≤ C n αn (t − s)−1 ϕn−1 (

t > s > τ;

t−s ), s−τ

(7.65)

t > s > τ;

t − t¯ t¯ − s t − t¯ |Un (t, s) − Un (t¯, s)| ≤ C n αn · · ϕn ( )ϕ0 ( ), t¯ − s s−τ t¯ − s

(7.66)

t > t¯ > s > τ ; (7.67)

t − t¯ t¯ − s t − t¯ )ϕn−1 ( )ϕ0 ( ), |Vn (t, s) − Vn (t¯, s)| ≤ C n αn (t¯ − s)−1 ϕ1 ( ¯ t−s s−τ t¯ − s for all t > t¯ > s > τ , where ϕ0 (a) = 1 + log(1 + 1/a), a > 0. Proof: (i) For n = 1, (7.66) and (7.68) are trivial; with (7.30) we get |U1 (t, s)| ≤ |Bε (τ1 )||T0 (s − τ ) − T0 (t − τ )| ≤ α1 N log(1 +

t−s ), s−τ

and t − t¯ ); |U1 (t, s) − U1 (t¯, s)| ≤ α1 |T0 (t − τ ) − T0 (t¯ − τ )| ≤ α1 N log(1 + t¯ − τ

(7.68)

206


with (7.64) log(1 +

t − t¯ t − t¯ t − t¯ (t¯ − s)/(s − τ ) t − t¯ t¯ − s )≤ = · ≤ log(1 + ), ¯ ¯ ¯ ¯ ¯ t−τ t−τ t − s 1 + (t − s)/(s − τ ) t−s s−τ

hence the assertion holds for n = 1 since ϕ0 ≥ 1. (ii) Let (7.65)∼(7.68) be satisfied for n. Then (7.59) and (7.30) yield |Vn+1 (t, s)|

≤ |Bε (τn+1 )| · {N (t − s)−1 αn C n ϕn ( t−s )} s−τ t−s ≤ αn+1 2N C n (t − s)−1 ϕn ( ) s−τ

t−s ) s−τ

+N C n αn (t − s)−1 ϕn−1 (

since ϕn−1 (a) ≤ ϕn (a) by Claim 4 (ii). Furthermore, with ϕ0 (a) = 1 + log(1 + 1/a) we get |Vn+1 (t, s) − Vn+1 (t¯, s)| ≤ |Bε (τn+1 )|{|(T˙0 (t − s) − T˙0 (t¯ − s))Un (t¯, s)| +|T˙0 (t − s)| · |Un (t, s) − Un (t¯, s)| + |T0 (t − s) − T0 (t¯ − s)||Vn (t¯, s)| +|T0 (t − s)||Vn (t, s) − Vn (t¯, s)|} t − t¯ t¯ − s t − t¯ ≤ αn+1 C n {2N ϕn ( )ϕ0 ( ) (t − s)(t¯ − s) s−τ t¯ − s t − t¯ t¯ − s t − t¯ )ϕn−1 ( ) · (1 + ϕ0 ( )) +N (t¯ − s)−1 ϕ1 ( ¯ t−s s−τ t¯ − s t¯ − s t − t¯ t − t¯ ≤ αn+1 4N C n (t¯ − s)−1 ϕn ( )ϕ0 ( )ϕ1 ( ), ¯ ¯ s−τ t−s t−s hence (7.66) and (7.68) hold for n + 1 ; here we used (7.64) as t − t¯ t − t¯ (t − t¯)/(t¯ − s) t − t¯ t − t¯ = = ≤ log(1 + ) = ϕ1 ( ), t−s t − t¯ + t¯ − s 1 + (t − t¯)/(t¯ − s) t¯ − s t¯ − s and Claim 4 (ii). (iii) For |Un+1 (t, s)| we obtain |Un+1 (t, s)| ≤ |Bε (τn+1 )|{|T0 (s − τ ) − T0 (t − τ )||Wn (s)| +|T0 (t − τ )||Wn (t) − Wn (s)| s (|T˙0 (s − r) − T˙0 (t − r)||Un (s, r)| + |T˙0 (t − r)||Un (s, r) − Un (t, r)|)dr + τ s + (|T0 (s − r) − T0 (t − r)||Vn (s, r)| + |T0 (t − r)||Vn (s, r) − Vn (t, r)|)dr} τ s t−s t−s s−r t−s n ≤ αn+1 C {2N ϕ1 ( ) + 2N · ϕn ( )ϕ0 ( )dr s−τ (s − r)(t − r) r − τ s −r τ s t−s s−r t−s +2N )(s − r)−1 ϕn−1 ( )ϕ0 ( )dr} ϕ1 ( s − r r − τ s −r τ


207

t−s ≤ αn+1 C n {2N ϕ1 ( ) s−τ s t−s s−r t−s +4N (s − r)−1 ϕ1 ( )ϕn ( )ϕ0 ( )dr}, s − r r − τ s −r τ since

t−s t−s ≤ ϕ1 ( ) t−r s−r

by (7.64). The transformation r = s − σ(s − τ ) now yields

s τ

(s − r)−1 ϕ1 (

t−s s−r t−s )ϕn ( )ϕ0 ( )dr s−r r−τ s−r

1

σ t − s −1 s−τ dσ ) log(1 + σ )[1 + log(1 + σ)] 1−σ s−τ t−s σ 0 t−s = ϕn+1 ( ), s−τ =

ϕn (

hence by Claim 4 (ii) we obtain (7.67) for n + 1. (iv) Similarly we get |Un+1 (t, s) − Un+1 (t¯, s)| ≤ |Bε (τn+1 )|{|T0 (t − τ ) − T0 (t¯ − τ )||Wn (t)| + |T0 (t¯ − τ )||Wn (t) − Wn (t¯)| s + (|T˙0 (t − r) − T˙0 (t¯ − r)||Un (t¯, r)| + |T˙0 (t − r)||Un (t, r) − Un (t¯, r)|)dr τ s + (|T0 (t − r) − T0 (t¯ − r)||Vn (t¯, r)| + |T0 (t − r)||Vn (t, r) − Vn (t¯, r)|)dr} τ s t − t¯ t − t¯ t¯ − r t¯ − t¯ n ≤ αn+1 C {2N ϕ1 ( ) + 2N ϕn ( )ϕ0 ( )dr ¯ t¯ − τ r−τ t¯ − r τ (t − r)(t − r) s t − t¯ ¯ t¯ − r t − t¯ +N ϕ1 ( )(t − r)−1 ϕn−1 ( )[1 + ϕ0 ( )]dr} t¯ − r r−τ t¯ − r τ t − t¯ ≤ αn+1 C n {2N ϕ1 ( ) ¯ t−τ s t − t¯ t¯ − r t − t¯ +4N ϕ1 ( )(t¯ − r)−1 ϕn ( )ϕ0 ( )dr}, ¯ ¯ t − r r − τ t −r τ with (t − t¯)/(t − r) ≤ ϕ1 ((t − t¯)/(t¯ − r)) by (7.64). The first term is okay by the estimate ϕ1 (

t − t¯ t − t¯ t − t¯ t¯ − s t − t¯ t¯ − s )≤ = · ≤ ϕ1 ( ). t¯ − τ t¯ − τ t¯ − s 1 + (t¯ − s)/(s − τ ) t¯ − s s−τ

208


The transformation r = τ + (s − τ )σ yields s t − t¯ ¯ t¯ − r t − t¯ ϕ1 ( )(t − r)−1 ϕn ( )ϕ0 ( )dr ¯ ¯ t−r r−τ t−r τ 1 a+1−σ b b dσ = ϕn ( )ϕ1 ( )ϕ0 ( ) = q, σ a + 1 − σ a + 1 − σ a + 1−σ 0 b a where a = (t¯−s)/(s−τ ) and b = (t− t¯)/(s−τ ). With ϕ0 ( a+1−σ ) ≤ ϕ0 ( ab )ϕ0 ( 1−σ ), (7.64) and Claim 4 (iv) imply 1 b a+1−σ b dσ q ≤ ϕn ( )ϕ0 ( ) a + 1 − σ σ a + 1 − σ a + 1−σ 0 1 b 1−σ b a dσ ≤ ϕn ( )ϕ0 ( ) · a 0 σ a+1−σ a+1−σ 1−σ b 1 1−σ a b dσ ≤ ϕn ( )ϕ1 ( )ϕ0 ( ) a 0 σ 1−σ a+1−σ 1−σ 1 b b 1−σ a a dσ ≤ ϕ0 ( ) ϕn ( )ϕ1 ( )ϕ0 ( ) a a 0 σ 1−σ 1−σ 1−σ t − t¯ t¯ − s t − t¯ = ϕn+1 ( )ϕ0 ( ), t¯ − s s−τ t¯ − s

and from this (7.67) for n + 1 follows. 2

Returning to Rn (t), Claim 5 yields the desired estimate. Claim 6 For each n ∈ ⺞ we have with η = η(ω) = |Rn (t)| ≤ C n+1 η n eωt

1 0

ϕn (

t0 0

e−ωτ db(τ )

1 − σ dσ ) . σ 1−σ

(7.69)

Proof: (7.61), (7.65) and (7.66) yield by means of (7.56) t t−s n |Zn (t)| ≤ N C αn {2 (t − s)−1 ϕn ( )ds + 1} s−τ τ 1 1 1 − σ dσ 1 − σ dσ ≤ C n N αn {1 + 2 ϕn ( ) } ≤ 3N C n αn ϕn ( ) σ 1−σ σ 1−σ 0 0 and therefore integration over τ1 , . . . , τn implies (7.69). 2

Thus it remains to derive estimates for the integrals over ϕn in (7.69). For this purpose we consider the singular integral equation 1 t s ds s ) log(1 + )[1 + log(1 + )] , t > 0. (7.70) ϕ(t) = log(1 + t) + ρ ϕ( 1 − s s t s 0


209

√ Suppose ϕ ∈ C(⺢+ , ⺢+ ) is a solutionof (7.70) such that ϕ(t) ≤ c t. Then it is n j easy to see by induction that ϕ(t) ≥ 0 ρ ϕj (t) for each n ∈ ⺞, hence 1 1 1 ∞ ∞ s ds s ds s ds ρn ϕn ( ) = ρn ϕn ( ) ≤ ϕ( ) 1−s s 1−s s 1−s s 0 0 0 0 0 1 ≤ c s−1/2 (1 − s)−1/2 ds = cπ < ∞, 0

i.e. with ρ = Cη we obtain by (7.69) |Tε (t)| ≤

∞

|Rn (t)| ≤ Ce

0

ωt

∞

n

ρ

0

n=0

1

ϕn (

s ds ) ≤ Ccπeωt , 1−s s

(7.71)

for all t ≥ 0, ε > 0, provided η is sufficiently small, i.e. ω sufficiently large. This is the uniform estimate for Tε (t). Hence it remains to prove Claim 7 Let ρ > 0 be sufficiently small. Then (7.70) has a solution ϕ ∈ C(⺢+ , ⺢+ ) such that √ ϕ(t) ≤ c t for t ≥ 0 and some c > 0. (7.72) Proof: Let Z denote the Banach space of all functions ψ ∈ C(⺢+ , ⺢) such that (7.72) holds, with norm ||ψ|| = supt≥0 |ψ(t)|t−1/2 and L : Z → Z the linear operator 1 s t ds (Lψ)(t) = ψ( )κ( ) , t ≥ 0, 1−s s s 0 where κ(a) = log(1 + a)[1 + log(1 + 1/a)], a > 0. Since ϕ1 (t) = log(1 + t) belongs to Z, we only have to show that L : Z → Z is bounded. For this purpose let ψ ∈ Z be fixed. Then we have 1 1 s t ds t (Lψ)(t) ≤ |ψ( )|κ( ) ≤ ||ψ|| s−1/2 (1 − s)−1/2 κ( )ds; 1−s s s s 0 0 with (7.64) one gets √ 4 22 √ κ(a) ≤ 2 a + a3/4 · 4a−1/4 = a, 3 3

a > 0,

hence 1

1 t 22 √ ds 44 √ √ s−1/2 (1 − s)−1/2 κ( )ds ≤ t (1 − s)−1/2 = 2 t, s 3 s 3 1/2 1/2

On the other hand, the change of variables a = t/s yields 1/2 ∞ √ t s−1/2 (1 − s)−1/2 κ( )ds = t κ(a)a−3/2 (1 − t/a)−1/2 da s 0 2t ∞ √√ √ ≤ t 2 κ(a)a−3/2 da ≤ c t, t > 0, 0

since the last integral is finite. Therefore, L : Z → Z is bounded. 2

t > 0.

210


Finally, we prove the continuity properties (ii) and (iii) asserted in Theorem 7.6. The arguments for S(t) in B(X) and B(Y ), resp. for T (t) in B(X, Y ) are again similar, and so we restrict attention to T (t). Continuity of T (t) in B(X, Y ) follows from the Neumann Series (7.50) for Tε (t). Since we know already that this series is absolutely and uniformly convergent w.r.t. t ≥ 0 and ε > 0, it suffices to show that each term Rnε (t) is B(X, Y )-continuous, uniformly w.r.t. ε > 0. This can be achieved by means of the estimates derived above. In fact, observe first that the decomposition t Zn (t) − Zn (t¯) = [T˙0 (t − s)Un (t, s)ds + T0 (t − s)Vn (t, s)]ds

t¯

t¯

[(T˙0 (t − s) − T˙0 (t¯ − s))Un (t¯, s) + T˙0 (t − s)(Un (t, s) − Un (t¯, s))]ds

+ τ

t¯

[T0 (t − s) − T0 (t¯ − s))Vn (t¯, s)] + T0 (t − s)(Vn (t, s) − Vn (t¯, s))]ds

+ τ

+(T0 (t − τ ) − T0 (t¯ − τ ))Wn (t) + T0 (t¯ − τ )(Wn (t) − Wn (t¯))

for t > t¯ > τ implies by Claim 4 and Claim 5 t t−s t − t¯ n ¯ |Zn (t) − Zn (t)| ≤ 2N αn C { (t − s)−1 ϕn ( )ds + ϕ1 ( ) ¯ s−τ t−τ t¯ t¯ t − t¯ t¯ − s t − t¯ + ϕn ( )ϕ0 ( )ds ¯ ¯ s−τ t−s τ (t − s)(t − s) t¯ t − t¯ t¯ − s t − t¯ + (t¯ − s)−1 ϕ1 ( )ϕn−1 ( )ϕ0 ( )ds} t¯ − s s−τ t¯ − s τ t − t¯ ≤ 8N αn C n ϕn+1 ( ). t¯ − τ Therefore, by integration we obtain ¯ Rnε (t) − Rnε (t) = Zn (t)dτ1 . . . dτn − Zn (t¯)dτ1 . . . dτn Dt Dt¯ = Zn (t)dτ1 . . . dτn + (Zn (t) − Zn (t¯))dτ1 . . . dτn , Dt \Dt¯

n

Dt¯

where Dt = {(τ1 , . . . , τn ) : 0 ≤ τj ≤ t − j+1 τj , j = 1, . . . , n}. Since Zn (t) is bounded by C n+1 αn · Cn for some constant Cn by Claim 6, this yields |Rnε (t) − Rnε (t¯)| = Cn C n+1 { αn dτ1 . . . dτn

Dt \Dt¯

t − t¯ )dτ1 . . . dτn + + αn ϕn+1 ( αn dτ1 . . . dτn } t¯ − τ Dt¯−h Dt¯\Dt¯−h t − t¯ ≤ Cn C n+1 {bn (t) − bn (t¯) + ϕn+1 ( )b(t0 )n + (bn (t¯) − bn (t¯ − h))}, h


211

where bn (t) = (db)∗n−1 ∗ b(t). This estimate shows that Rnε (t) is left-continuous in B(X, Y ) and right-continuous in B(X, Y ) at each point of continuity of bn (t), uniformly w.r.t. ε > 0. Since Rnε → Rn in L1loc (⺢+ , B(X, Y ) this implies that Rn (t) has the same continuity properties and therefore T (t) is left-continuous in B(X, Y ) and right-continuous in B(X, Y ) at each point of continuity of the resolvent kernel r of b, i.e. of the solution of r = b + db ∗ r. This completes the proof of Theorem 7.6. 2

7.7 Comments a) Analytic resolvents for equations of nonscalar type have been studied for the first time by Grimmer and Pritchard [144]. Their results have been improved for first order equations with main part, i.e. t u(t) ˙ = Au(t) + dB(t − s)u(s) + f (t), t ∈ J, (7.73) 0

where A generates an analytic semigroup in X and B ∈ BVloc (⺢+ ; B(XA , X)), by Da Prato and Iannelli [68], Lunardi [224], Pr¨ uss [271], [273], and Tanabe [322]. b) The main result of Section 7.2 is new. It is an extension of the approach presented in Section 3 for equations of salar type. c) Theorem 7.3 for equations of variational type seems also to be new. It appears to be possible to replace the assumption of uniform measurability by mere weak measurability, however, the proof then cannot be based on Theorems 7.1 and 7.2, it must be carried through directly. For the variational approach to partial differential equations there is an enormous literature; let us mention only the monographs of Lions [215] and Showalter [303]. d) Theorems 7.4 and 7.6 are extensions of results of Pr¨ uss [271], [273] where the first order case with main part, i.e. (7.73), was considered. Friedman and Shinbrot [123] is probably the earliest paper which considers (7.73) even in the nonautonomous case, however only solvability for initial values u0 ∈ D(Aα ) for some α > 0 was obtained under additional regularity assumptions on B(t). In Pr¨ uss [270], and Tanabe [320], [321] the nonautonomous resolvent has been constructed. Grimmer and Kappel [140] used perturbation series in the frequency domain to establish ˆ existence of the resolvent for (7.73) assuming a certain decay of B(λ). This decay requires additional regularity of B(t), but implies an improved behaviour of the resolvent. Lunardi [225] contains maximal regularity results for nonautonomous parabolic first order problems.

8

Parabolic Problems in Lp -Spaces

The subject of this section is the Lp -theory for parabolic equations with main part. The first three subsections prepare the approach via sums of commuting linear operators; the two basic results, i.e. a vector-valued Fourier-multiplier theorem and the Dore-Venni theorem, are stated without proof. After a thorough study of Volterra operators in Lp (⺢; X), these results are then applied to prove maximal regularity for parabolic equations with main part in Lp (J; X), where J ⊂ ⺢+ is a closed interval, X belongs to the class HT introduced in Section 8.2, and p ∈ (1, ∞). 8.1 Operators with Bounded Imaginary Powers Let X be a complex Banach space, and A a closed linear operator in X with dense domain D(A), such that (SL1) N (A) = {0}, R(A) = X, (−∞, 0) ⊂ ρ(A); (SL2) |t(t + A)−1 | ≤ M for all t > 0, and some M < ∞; are satisfied. Operators satisfying (SL1) and (SL2) will be called sectorial, and S(X) denotes the class of sectorial operators in X. Observe that A ∈ S(X) implies A−1 ∈ S(X), and cA ∈ S(X), for each c > 0. Also, if A ∈ S(X) and T ∈ B(X, Z) is invertible then T AT −1 ∈ S(Z). If A ∈ S(X), then t(t + A)−1 → I strongly as t → ∞, and t(t + A)−1 → 0 strongly as t → 0+. For any A ∈ S(X) one can define complex powers Az , where z ∈ ⺓ is arbitrary; this will be done as follows. Assume first that A and A−1 are both bounded. Choose a piecewise smooth simple closed path Γ, surrounding the spectrum σ(A) of A counterclockwise, which does not intersect the negative real axis (−∞, 0]. Then define Az by means of the Dunford integral 1 Az = λz (λ − A)−1 dλ, z ∈ ⺓, (8.1) 2πi Γ where λz = exp(z log λ) = exp(z log |λ| + iz arg λ) denotes the principal branch of the multivalued function λz . Obviously, Az ∈ B(X) for each z ∈ ⺓, the mapping ⺓ z → Az ∈ B(X) is holomorphic, and {Az : z ∈ ⺓} ⊂ B(X) has the group property Az Aw = Az+w , A0 = I, for all z, w ∈ ⺓, since the Dunford integral induces an algebra homomorphism. For 0 > Re z > −1 we may contract the contour Γ to the negative real axis to the result sin πz ∞ z Az = − t (t + A)−1 dt, −1 < Re z < 0. (8.2) π 0 Splitting the integral in (8.2) at t = 1 and using the identities t(t + A)−1 = I − (t + A)−1 A

and

1 1 (t + A)−1 = A−1 − (t + A)−1 A−1 , t t


212

8. Parabolic Problems in Lp

213

(8.2) becomes A

z

1 sin πz 1 1 −1 = { − A + tz+1 (t + A)−1 A−1 dt π z 1+z 0 ∞ + tz−1 (t + A)−1 Adt}.

(8.3)

1

This formula not only makes sense for −1 < Re z < 0, but by analytic continuation is valid for all |Re z| < 1. Let A ∈ S(X) be arbitrary now. Then (8.1) can no longer be used to define Az since the integral diverges at λ = 0 and also at λ = ∞, in general. However, we may use (8.3) to define Az x, x ∈ D(A) ∩ R(A), |Re z| < 1; the function f (z) = Az x is still holomorphic in this strip. The complex powers of A defined this way are densely defined and closable, since D(A) ∩ R(A) is dense in X, Az commutes with (t + A)−1 , and is bounded from D(A) ∩ R(A) equipped with its natural norm |x| + |Ax| + |A−1 x| to X. We denote the closure of Az defined by (8.3) on D(A) ∩ R(A) again by Az . Next we show that Az still has the group property. For this purpose we introduce the approximations Aε = (ε + A)(1 + εA)−1 ,

ε > 0,

of elements A ∈ S(X). Obviously, Aε is bounded and invertible, and (t + Aε )−1

= (t + (ε + A)(1 + εA)−1 )−1 = (1 + εA)(t + ε + (1 + εt)A)−1 1 t+ε = (1 + εA)( + A)−1 , 1 + εt 1 + εt

t, ε > 0,

i.e. ρ(Aε ) ⊃ (−∞, 0], and as ε → 0, (t + Aε )−1 → (t + A)−1 in B(X) for each t > 0, −1 Aε x → Ax for each x ∈ D(A), A−1 x for each x ∈ R(A). Since ε x →A |t(t + Aε )−1 | ≤

Mt εt(M + 1) + ≤ 2M + 1, t+ε 1 + εt

t, ε > 0,

we have Aε ∈ S(X) for each ε > 0, and there is a constant M for (SL2) which is independent of ε. From (8.3) it is now evident that Azε x →ε→0 Az x,

|Re z| < 1, x ∈ D(A) ∩ R(A),

holds. The group property for Azε then implies the group property for Az . Let us summarize what we have proved so far in Proposition 8.1 Suppose A ∈ S(X), let Az be defined by (8.3) for |Re z| < 1. Then (i) Az x is holomorphic on |Re z| < 1, for each x ∈ D(A) ∩ R(A); (ii) Az is closable for each |Re z| < 1;

214


(iii) Az+w x = Az Aw x for Re z, Re w, Re(z + w) ∈ (−1, 1), x ∈ D(A) ∩ R(A); (iv) Aε = (ε + A)(1 + εA)−1 is bounded, sectorial, and invertible, for each ε > 0. (v) Az x = limε→0 Azε x, x ∈ D(A) ∩ R(A), |Re z| < 1. Proposition 8.1 shows that the following definition makes sense. Definition 8.1 Suppose A ∈ S(X). Then A is said to admit bounded imaginary powers if Ais ∈ B(X) for each s ∈ ⺢, and there is a constant C > 0 such that |Ais | ≤ C for |s| ≤ 1. The class of such operators will be denoted by BIP(X). Since by Proposition 8.1, Ais has the group property, it is clear that A admits bounded imaginary powers if and only if {Ais : s ∈ ⺢} forms a strongly continuous group of bounded linear operators in X. The growth bound θA of this group, i.e. θA = lim|s|→∞

1 log |Ais | |s|

(8.4)

will be called the power angle of A. This is motivated by the fact that σ(A) ⊂ Σ(0, θA ) holds; see Pr¨ uss and Sohr [283]. It is in general not easy to verify that a given A ∈ S(X) belongs to BIP(X), although quite a few classes of operators are known for which the answer is positive. In Section 8.4 it will be shown that Volterra operators in Lp (J; X) admit bounded imaginary powers if the kernels are nice enough. For further results and remarks on the class BIP(X) see the comments given in Section 8.7. For a first application consider the fractional power spaces Xα = XAα = (D(Aα ), | · |α ), |x|α = |x| + |Aα x|,

0 < α < 1,

where A ∈ S(X); the embeddings XA → Xβ → Xα → X,

1 > β > α > 0,

are well known. If A belongs to BIP(X), a characterization of Xα in terms of complex interpolation spaces can be derived. Theorem 8.1 Suppose A ∈ BIP(X). Then Xα ∼ = [X, XA ]α ,

α ∈ (0, 1),

(8.5)

the complex interpolation space between X and XA → X of order α. For a proof we refer to Triebel [328], pp. 103-104, or Yagi [342]. The importance of Theorem 8.1 is twofold. It shows on one hand that Xα is largely independent of A; for instance if A, B ∈ BIP(X) are such that D(A) = D(B) then D(Aα ) = D(B α ) for all α ∈ (0, 1). On the other hand, (8.5) makes the tools of complex interpolation theory available for fractional power spaces and it becomes possible to compute


215

Xα in many cases; see also Section 8.4 below. For example, the reiteration theorem yields the relation [Xα , Xβ ]θ = Xα(1−θ)+θβ ,

for all 0 ≤ α < β ≤ 1, θ ∈ (0, 1).

for complex interpolation of fractional power spaces of operators A ∈ BIP(X). 8.2 Vector-Valued Multiplier Theorems One of the most important tools to prove that translation invariant operators on Lp -spaces admit bounded imaginary powers are Fourier multiplier theorems. Let us recall first the Marcienkiewicz multiplier theorem for Lp (⺢), 1 < p < ∞. Marcienkiewicz Multiplier Theorem Suppose m ∈ BVloc (⺢ \ {0}) satisfies 1 R>0 2R

||m|| := |m|∞ + sup

R −R

|ρ||dm(ρ)| < ∞.

(8.6)

Then there is an operator Tm ∈ B(Lp (⺢)), 1 < p < ∞, such that (Tm f )∼ = m(ρ)f˜(ρ),

for ρ ∈ ⺢, f˜ ∈ C0∞ (⺢ \ {0}),

(8.7)

and |Tm |B(Lp (⺢)) ≤ Cp ||m||,

(8.8)

where Cp denotes a constant only depending on p ∈ (1, ∞). Note that the set of all f ∈ Lp (⺢) such that f˜ ∈ C0∞ (⺢ \ {0}) is dense in Lp (⺢); therefore (8.7) uniquely determines Tm . For many applications it is enough to check the Mikhlin multiplier condition 1,∞ m ∈ Wloc (⺢ \ {0}),

ess sup{|m(ρ)| + |ρm (ρ)| : ρ ∈ ⺢ \ {0}} < ∞,

(8.9)

which obviously implies (8.6). Let us illustrate the connection between multiplier theorems and imaginary powers of translation invariant operators on Lp (⺢) by means of the following instructive example. Example 8.1 Let X = Lp (⺢), 1 ≤ p < ∞, A = d/dt, D(A) = W 1,p (⺢); A clearly is the generator of the C0 -group of translations defined by (T (τ )f )(t) = f (t + τ ), t, τ ∈ ⺢. It is not difficult to compute the Fourier multipliers, the symbols associated with A, T (τ ), (λ − A)−1 , etc. T (τ ) ←→ eiτ ρ , τ ∈ ⺢; A ←→ iρ; is Ais ε ←→ [aε (ρ)] , ε > 0, s ∈ ⺢;

(λ − A)−1 ←→ (λ − iρ)−1 , Re λ = 0; ε + iρ , ε > 0; Aε ←→ aε (ρ) = 1 + iερ Ais ←→ (iρ)is , s ∈ ⺢.

216


Thus the symbol associated with Ais is given by ms (ρ) = (iρ)is . Since m s (ρ) = isms (ρ)/ρ, ρ = 0, it is not difficult to check the Mikhlin condition (8.9). |ms (ρ)| + |ρm s (ρ)|

= (1 + |s|)|ms (ρ)| = (1 + |s|) exp(−s ≤ (1 + |s|)e 2 |s| , π

π sign ρ) 2

s ∈ ⺢.

By the Marcienkiewicz multiplier theorem we therefore obtain A ∈ BIP(Lp (⺢)) and π |Ais |B(Lp (⺢)) ≤ Cp (1 + |s|)e 2 |s| , s ∈ ⺢, p ∈ (1, ∞). (8.10) On the other hand, A ∈ BIP(L1 (⺢)), since it is well known that m(ρ) is a multi plier for L1 (⺢) if and only if m(ρ) = dk(ρ), ρ ∈ ⺢, for some function k ∈ BV (⺢); in particular, m(ρ) must be uniformly continuous on ⺢, which is not the case for ms (ρ). 2

Turning to the vector-valued case now, it is obvious that Example 8.1 carries over to Lp (⺢; X), X a Banach space, whenever the Marcienkiewicz multiplier theorem holds for Lp (⺢; X). One of the simplest multipliers which satisfy (8.6) or (8.9) is the symbol of the Hilbert transform h(ρ) = −i sgn ρ,

ρ = 0,

(8.11)

and therefore the Banach space X must be subject to the property that the Hilbert transform defined by 1 (Hf )(t) = lim f (t − s)ds/s, t ∈ ⺢, (8.12) ε→0 π ε≤|s|≤R R→∞

is bounded in Lp (⺢; X) for some p ∈ (1, ∞); the limit in (8.12) is to be understood in the Lp -sense. Note that (8.12) exists in Lp (⺢; X) for any Banach space X, p ∈ (1, ∞) if f˜ ∈ C0∞ (⺢ \ {0}; X). The class of Banach spaces X such that the Hilbert transform H is bounded in Lp (⺢; X) will be denoted by HT . Even more is true. Suppose A = d/dt belongs to BIP(Lp (⺢; X)) for some p ∈ (1, ∞). Then −A ∈ BIP(Lp (⺢; X)) as well, since −A is similar to A with similarity (Rf )(t) = f (−t), t ∈ ⺢, the reflection at 0, which induces an isomorphism on Lp (⺢; X). Consider now the operator K = (−i/ sinh π)[A−i (−A)i − cosh π] ∈ B(Lp (⺢; X)); its symbol is given by k(ρ) = −(i/ sinh π)[(iρ)−i (−iρ)i − cosh π] π π = −(i/ sinh π)[exp(−i log |ρ| + sgnρ) · exp(i log |ρ| + sgnρ) − cosh π] 2 2 = −i · (exp(πsgnρ) − cosh π)/ sinh π = −isgnρ, ρ = 0, i.e. k(ρ) = h(ρ). Thus if A = d/dt ∈ BIP(Lp (⺢; X)) then X must belong to the class HT .


217

It is surprising that the Marcienkiewicz theorem is valid in the vector-valued case if and only if A = d/dt ∈ BIP(Lp (⺢; X)) for some p ∈ (1, ∞), and if and only if the Banach space X belongs to the class HT . Theorem 8.2 Suppose X belongs to the class HT , and that m ∈ BVloc (⺢ \ {0}) satisfies (8.6). Then the operator Tm , uniquely defined by (Tm f )∼ (ρ) = m(ρ)f˜(ρ),

ρ ∈ ⺢, f˜ ∈ C0∞ (⺢ \ {0}; X),

(8.13)

is bounded on Lp (⺢; X), 1 < p < ∞, and |Tm |B(Lp (⺢;X)) ≤ Cp (X)||m||,

(8.14)

where Cp (X) denotes a constant only depending on p and X. For the proof we refer to Bourgain [32] and Zimmermann [349]]. The basic reference for the class HT is the survey article Burkholder [35], where also two other characterizations for the class HT are given, a probabilistic one, and a geometrical one. To describe the latter, recall that a Banach space X is termed ζ-convex, if there is a function ζ : X × X → ⺢ which is convex in each of its variables and such that ζ(0, 0) > 0 and ζ(x, y) ≤ |x + y|

for all x, y ∈ X with |x| = |y| = 1.

A Banach space X belongs to the class HT if and only if X is ζ-convex. See also Section 8.7 for comments on further properties of the class HT . 8.3 Sums of Commuting Linear Operators Let X be a Banach space, A, B closed linear operators in X, and let A + B be defined by (A + B)x = Ax + Bx,

x ∈ D(A + B) = D(A) ∩ D(B).

(8.15)

If A + B is closed and ρ(A + B) = ∅, then for each λ ∈ ρ(A + B), the solution x of (A + B)x = λx + y

(8.16)

belongs to D(A) ∩ D(B); consequently it has the regularity induced by A as well as that coming from B, it has maximal regularity. On the other hand, if A + B is closable but not closed, and λ ∈ ρ(A + B) then (8.16) only admits generalized solutions in the sense that there are sequences (xn ) ⊂ D(A) ∩ D(B), xn → x, and yn → y such that Axn + Bxn = λxn + yn , n ∈ ⺞. In general, nothing can be said on A + B, it need not even be closable, unless very restrictive assumptions on A and B are imposed.

218


Two closed linear operators A, B in X are said to commute if there are λ0 ∈ ρ(A), μ0 ∈ ρ(B) such that (λ0 − A)−1 (μ0 − B)−1 = (μ0 − B)−1 (λ0 − A)−1 .

(8.17)

The spectral angle φA ∈ [0, π] of A ∈ S(X) is defined by φA = inf{φ : ρ(−A) ⊃ Σ(0, π − φ), |λ(λ + A)−1 | ≤ Cφ on Σ(0, π − φ)}. (8.18) It is obvious that σ(A) ⊂ Σ(0, φA ) holds, and for A ∈ BIP(X) we have the relation φA ≤ θA ; see Pr¨ uss and Sohr [283]. We are now in position to state the following fundamental result which is due to Da Prato and Grisvard [64]. Theorem 8.3 Suppose A, B are closed linear operators in X which commute and satisfy the parabolicity condition φA + φB < π. Then (i) A + B is closable; (ii) L = A + B ∈ S(X) has spectral angle φL ≤ max{φA , φB }; (iii) D(L) ⊂ DA (1, ∞) ∩ DB (1, ∞); (iv) If A or B is invertible, then L is also invertible; (v) y ∈ DA (α, q) [or y ∈ DB (α, q)] and λ ∈ Σ(0, π − φL ) then x = (λ + L)−1 y ∈ D(A) ∩ B(B) and Ax, Bx ∈ DA (α, q) [or y ∈ DB (α, q)], for any α ∈ (0, 1) and q ∈ [1, ∞]. Here DA (α, q) = (X, XA )α,q denote the real interpolation spaces between X and XA whenever A ∈ S(X), and DA (1, ∞) = {x ∈ X : ess sup |tA2 (t + A)−2 x| < ∞}; observe D(A) ⊂ DA (1, ∞) ⊂ DA (α, q), for all α ∈ (0, 1) and q ∈ [1, ∞]. There are examples showing that even in a Hilbert space, A + B need not be closed, in the situation of Theorem 8.3; cp. Baillon and Clément [16]. However, if the assumptions on A, B and X are strengthened, one can prove the following result which extends a remarkable recent theorem due to Dore and Venni [101], [102]. Theorem 8.4 Suppose X belongs to the class HT , and assume A, B ∈ BIP(X) commute and satisfy the strong parabolicity condition θA + θB < π. Then (i) A + B is closed and sectorial; (ii) A + B ∈ BIP(X) with θA+B ≤ max{θA , θB }; (iii) there is a constant C > 0 such that |Ax| + |Bx| ≤ C|Ax + Bx|,

x ∈ D(A) ∩ D(B).

In particular, if A or B is invertible, then A + B is invertible as well. For a proof of this result we refer to Pr¨ uss and Sohr [283].

(8.19)


219

Combining Theorem 8.4 with the methods introduced by Da Prato and Grisvard [64] we obtain the following result. Theorem 8.5 Suppose X belongs to the class HT , and assume (i) ωA + A, ωB + B ∈ BIP(X), for some ωA , ωB ∈ ⺢; (ii) A and B commute; (iii) θA+ωA + θB+ωB < π. Then A + B is closed with domain D(A + B) = D(A) ∩ D(B) and σ(A + B) ⊂ σ(A) + σ(B). In particular, if σ(A) ∩ σ(−B) = ∅ then A + B is invertible. Proof: a) Theorem 8.4 applied to ωA + A and ωB + B shows that ωA + ωB + A + B is closed, hence A + B is closed. Suppose λ ∈ σ(A)+σ(B); based on ideas of da Prato and Grisvard [64], we will derive an explicit formula for (λ − A − B)−1 which then shows that λ ∈ σ(A + B). For this observe first that λ ∈ σ(A) + σ(B) is equivalent to σ(A) ∩ σ(λ − B) = ∅. On the other hand, the assumptions on A and B imply σ(A) ⊂ Σ(−ωA , θωA +A ), σ(λ − B) ⊂ −Σ(−λ − ωB , θωB +B ) and we may choose θωB +B < θ < π − θωA +A such that M1 for all z ∈ Σ(ωA , θ), (8.20) |(z + A)−1 | ≤ |z − ωA | as well as |(z − (λ − B))−1 | ≤

M2 |z − ωB − λ|

for all z ∈ Σ(ωB + λ, θ).

(8.21)

w.l.o.g. we may assume θωB +B < π/2 and then also θ < π/2. iθ −iθ b) Consider √ the oriented rays Γ−∞ = −ωA −(∞, R]e , Γ∞ = −ωA −[R, ∞]e , where R > 2|λ+ωA +ωB | and so large that Γ−∞ ∪Γ∞ ⊂ Σ(ωB +λ, θ). By virtue of (iii) and since σ(A) ∩ σ(λ − B) = ∅ it is apparent that r0 = d(σ(A), σ(λ − B))/4 > 0. Consider the open square QR with center at −ωA generated by the corners −ωa ± Re±iθ ; w.l.o.g. 0 ∈ QR . Choose a grid of mesh size less than r0 on QR , and denote the open squares generated by this grid by Qj , 1 ≤ j ≤ N ; w.l.o.g. we may assume 0 ∈ ∂Qj for all j. With J = {j : Qj ∩ σ(A) = ∅} define U0 = ∪j∈J Qj , and let Γ 0 = ∂U0 with orientation induced by negative orientation of the boundaries of the squares Qj ; observe that Γ 0 ∩ σ(B) = ∅. Let Γ+ R denote that part of ∂QR running from −ωA − Reiθ to −ωA Re−iθ which is contained in Σ(ωB + λ, θ) and + Γ− R = ∂QR \ΓR with positive orientation; c.p. Figure 8.1.

If Γ0 ∩ ∂QR = ∅ we let Γ0 = Γ+ R ; otherwise Γ0 will defined as follows. Starting +

from −ωA −Reiθ move along Γ+ R until Γ0 is reached; then follow Γ0 until ΓR is met + again. Continuing this way, after finitely many switches between ΓR and Γ 0 one arrives at −ωA −Reiθ ; from the curve obtained this way remove all closed connected subarcs, and denote the remaining curve by Γ1 . By construction, Γ1 ∪ Γ− R then is a closed positively oriented simple curve which does not intersect σ(A). Therefore by the Jordan curve theorem, for each μ ∈ σ(A) we have either n(Γ1 ∪ Γ− R , μ) = 0 − or n(Γ1 ∪ Γ− R , μ) = 1 according to whether μ is outside or inside of Γ1 ∪ ΓR , here n denotes the winding number.

220


Let U1 denote the union of all Qj such that Qj ∩ σ(A) ∩ {μ : n(Γ1 ∪ Γ− R , μ) = 1} = ∅, and let Γ2 = ∂U1 with negative orientation; then we have Γ2 ∩ σ(A) = ∅ and n(Γ1 ∪ Γ− R ∪ Γ2 , μ) = 0 for all μ ∈ σ(A). Similarly, for each μ ∈ σ(λ − B) ∩ QR we have either n(Γ1 ∪ Γ− R ∪ Γ2 , μ) = 0 or n(Γ1 ∪ Γ− R ∪ Γ2 , μ) = 1. Denoting by U2 the union of all Qj such that Qj ∩σ(λ−B)∩{μ : n(Γ1 ∪Γ− R ∪Γ2 , μ) = 0} = ∅, Γ3 = ∂U2 with positive orientation, − we obtain n(Γ1 ∪ −Γ+ ∪ Γ 2 ∪ Γ3 , μ) = n(Γ1 ∪ ΓR ∪ Γ2 , μ) + n(Γ3 ; μ) − 1 = 0 for all R μ ∈ σ(λ − B); Finally, we let Γ = Γ−∞ ∪ Γ1 ∪ Γ∞ ∪ Γ2 ∪ Γ3 , note that n(Γ3 , μ) = 0 for all μ ∈ σ(A). Γ1 σ Α

σ λ−

2θ Α

Γ2

2θ Β Γ3

Γ2

Γ1

Figure 8.1:

c) Define Sλ =

1 2πi

Γ

Integration path for (8.22)

(z − λ + B)−1 (z − A)−1 dz;

(8.22)

Since the unbounded parts of Γ, i.e. the rays Γ∞ , Γ−∞ , are contained in Σ(ωB + λ, θ) ∩ −Σ(ωA , θ), with (8.20) and (8.21), the integral in (8.22) is absolutely convergent, and so Sλ ∈ B(X); note that Γ does not intersect σ(A) ∪ σ(λ − B). We claim that Sλ = (λ − (A + B))−1 ; by closedness of A + B for this it is sufficient to prove Sλ (λ − (A + B))x = x for each x ∈ D(A) ∩ D(B). (8.23) For z ∈ Γ we have (z − λ + B)−1 (z − A)−1 (λx − Ax − Bx) = (λ − B)(z − λ + B)−1 (z − A)−1 x − (z − λ + B)−1 A(z − A)−1 x = −(z − A)−1 x + (z − λ + B)−1 x = [−(z − A)−1 Ax + (z − (λ − B))−1 (λ − B)x]/z, hence Sλ (λ−(A+B))x = −

1 2πi

(z−A)−1 Axdz/z+ Γ

1 2πi

(z−(λ−B))−1 (λ−B)xdz/z. Γ


221

Contracting Γ into ΓA = Γ1 ∪ Γ− R ∪ Γ2 ∪ Γ3 within the sector −Σ(ωA , θ) in the first integral of the right hand side of this identity, and Γ into ΓB = Γ1 ∪(−Γ+ R )∪Γ2 ∪Γ3 within Σ(−ωA , π − θ) in the second one, by Cauchy’s theorem we obtain with n(ΓA , σ(A)) = 0 and n(ΓB , σ(λ − B)) = 0 by construction 1 Sλ (λ − (A + B))x = − (z − A)−1 Axdz/z 2πi ΓA 1 + (z − (λ − B))−1 (λ − B)xdz/z 2πi ΓB = [n(ΓA , 0) − n(ΓB , 0)]x = x, since ΓB = ΓA − ∂QR , i.e. n(ΓB , 0) = n(ΓA , 0) − 1; observe that 0 ∈ ΓA ∪ ΓB . Thus we have derived (8.23) and the proof is complete. 2

Observe that the commutativity assumption in Theorems 8.4 and 8.5 is not as severe as it is in Theorem 8.3, since a posteriori it may be relaxed by introducing a perturbation C : D(A) ∩ D(B) → X which is A + B-bounded with sufficiently small bound, i.e. |Cx| ≤ a|Ax| + b|Bx| + c|x|,

x ∈ D(A) ∩ D(B),

where a, b are sufficiently small. In the situation of Theorem 8.3 only perturbations of ‘lower order’ are possible, i.e. D(C) ⊃ DA (1, ∞) ∩ DB (1, ∞). We refer also to the comments in Section 8.7 where more remarks are given. 8.4 Volterra Operators in Lp Let X be a Banach space of class HT , and consider Volterra operators in the spaces Lp (⺢; X), 1 < p < ∞, of the form b ∗ u, where b ∈ L1loc (⺢+ ) is of subexponential growth. If b is θ-sectorial and 1-regular, we show that b gives rise to an operator B ∈ BIP(X); this will basically be a consequence of Theorem 8.2. We shall need the following result containing further properties of 1-regular kernels. Lemma 8.1 Suppose b ∈ L1loc (⺢+ ) is of subexponential growth and 1-regular. Then (i) ˆb(iρ) := limλ→iρ ˆb(λ) exists for each ρ = 0; (ii) ˆb(λ) = 0 for each Re λ ≥ 0, λ = 0; 1,∞ (iii) ˆb(i·) ∈ Wloc (⺢ \ {0});

ˆ ˆ (iv) |ρb (iρ)| ≤ c|b(iρ)| for a.a. ρ ∈ ⺢; (v) there is a constant c > 0 such that c|ˆb(|λ|)| ≤ |ˆb(λ)| ≤ c−1 |ˆb(|λ|)|, (iv) limr→∞ ˆb(reiϕ ) = 0 uniformly for |ϕ| ≤

π . 2

Re λ ≥ 0, λ = 0;

222


Proof: We already observed in Section 3.2 that 1-regular kernels do not admit zeros in ⺓+ . Therefore, by simple connectedness of ⺓+ , there is a holomorphic function g(λ) such that exp(g(λ)) = ˆb(λ) on ⺓+ . 1-regularity of ˆb(λ) then implies |g (λ)| = |ˆb (λ)/ˆb(λ)| ≤ c/|λ|,

Re λ > 0,

i.e. g (λ) is bounded on ⺓+ \ Bη (0), η > 0, arbitrary; hence g(iρ) = lim g(λ), λ→iρ

ρ = 0,

exists, even uniformly for |ρ| ≥ η. Therefore, ˆb(iρ) = lim ˆb(λ) = exp(g(iρ)), λ→iρ

ρ = 0,

also exists, and the convergence is uniform for 0 < η ≤ |ρ| ≤ η −1 ; this implies (i) and (ii), as well as the boundedness of ˆb(λ) on compact subsets of ⺓+ \ {0}. The estimate (3.7) of 1-regularity of b then shows that also ˆb (λ) is bounded on compact subsets, and so the nontangential limits ϕ(ρ) of ˆb (λ) exist a.e. on ⺢, cp. Duren [106], Theorem 1.3. From the identity

ρ

ˆb(σ + iρ) − ˆb(σ + iτ ) = i

ˆb (σ + is)ds,

σ > 0, ρ · τ > 0

τ

we then may conclude

ρ

ˆb(iρ) − ˆb(iτ ) = i

ϕ(s)ds,

ρ · τ > 0,

τ 1,∞ i.e. ˆb(i·) ∈ Wloc (⺢ \ {0}) and ϕ(s) = ˆb (iρ) a.e.; we have proved (iii). (iv) follows from 1-regularity of b. Finally, to prove (v) and (vi) we write

g(λ) = g(|λ|) + i

arg λ

g (|λ|eiϕ )|λ|eiϕ dϕ,

0

Re λ ≥ 0, λ = 0,

hence by 1-regularity of b(t) −cπ/2 + Re g(|λ|) ≤ Re g(λ) ≤ Re g(|λ|) + cπ/2 and therefore with |ˆb(λ)| = exp(Re g(λ)), (v) follows. (vi) is a direct consequence of (v) since b(λ) → 0 as λ → ∞. 2

We are now ready for the main result of this subsection. In the sequel, Lp (⺢+ ; X) is identified with a subspace of Lp (⺢; X), extending functions from Lp (⺢+ ; X) by 0 to (−∞, 0).


223

Theorem 8.6 Suppose X belongs to the class HT , p ∈ (1, ∞), and let b ∈ L1loc (⺢+ ) be of subexponential growth; assume that b is 1-regular and θ-sectorial, for some θ < π. Then there is a unique operator B ∈ S(Lp (⺢; X)) such that (Bf )∼ (ρ) =

1 ˜ f (ρ), ˆb(iρ)

ρ ∈ ⺢, f˜ ∈ C0∞ (⺢ \ {0}; X).

(8.24)

Moreover, B has the following properties: (i) B commutes with the group of translations; (ii) (μ + B)−1 Lp (⺢+ ; X) ⊂ Lp (⺢+ ; X) for each μ > 0, i.e. B is causal; (iii) B ∈ BIP(Lp (⺢; X)), and θB = φB = θb , where θb = sup{| arg ˆb(λ)| : Re λ > 0};

(8.25)

(iv) σ(B) = {1/ˆb(iρ) : ρ ∈ ⺢ \ {0}}. Proof: By Lemma 8.1, ˆb(λ) admits continuous extension to ⺓+ \ {0}, and the inequality | arg ˆb(iρ)| ≤ θb holds for all ρ = 0. Thus for each η > 0, there is a constant c(η) such that c(η)|1 + μˆb(λ)| ≥ 1 + |μˆb(λ)|,

λ ∈ ⺓+ \ {0}, | arg μ| ≤ π − θb − η = ϑη

holds. Define multipliers rμ (ρ) by means of rμ (ρ) =

ˆb(iρ) , 1 + μˆb(iρ)

ρ ∈ ⺢, ρ = 0, μ ∈ Σ(0, ϑη );

then |rμ (ρ)| ≤ c(η)

|ˆb(iρ)| c(η) ≤ , |μ| 1 + |μ||ˆb(iρ)|

and from rμ (ρ) =

ρ ∈ ⺢, ρ = 0, μ ∈ Σ(0, ϑη ),

iˆb (iρ) (1 + μˆb(iρ))2

using the 1-regularity of b(t) we obtain by Lemma 8.1 |ρrμ (ρ)| ≤ |ρ

ˆb (iρ) c(η)2 |ˆb(iρ)| c · c(η)2 |· ≤ , ˆb(iρ) (1 + |μ||ˆb(iρ)|)2 |μ|

ρ ∈ ⺢ \ {0}, μ ∈ Σ(0, ϑη ).

Therefore, by Theorem 8.2, there are operators Rμ ∈ B(Lp (⺢; X)), uniquely defined by ˜ R ρ ∈ ⺢, f˜ ∈ C0∞ (⺢ \ {0}; X), (8.26) μ f (ρ) = rμ (ρ)f (ρ), and they are subject to the estimate |Rμ | ≤

C(η) , |μ|

μ ∈ Σ(0, ϑη ).

(8.27)

224


The relation rμ (ρ) − rν (ρ) = (ν − μ)rμ (ρ)rν (ρ),

ρ ∈ ⺢, ρ = 0, μ, ν ∈ Σ(0, ϑη ),

then implies the identity Rμ − Rν = (ν − μ)Rμ Rν ,

μ, ν ∈ Σ(0, ϑη ).

(8.28)

Thus the family {Rμ }μ∈Σ(0,ϑη ) ⊂ B(Lp (⺢; X)) is a pseudo-resolvent in the sense of Hille and Phillips [180], p.185. According to Theorem 5.8.3 of [180], the kernels N (Rμ ) and the ranges R(Rμ ) are independent of μ, and there is a unique closed linear operator B in Lp (⺢; X) such that Rμ = (μ + B)−1 iff the kernel N (Rμ ) of Rμ is trivial. The operator B (if it exists) is densely defined iff R(Rμ ) is dense in Lp (⺢; X); in fact, B is given by B = Rμ−1 − μI, i.e. D(B) = R(Rμ ). Since ˆb(iρ) = 0 for ρ = 0 there follows μrμ (ρ) → 1 a.e. as μ → ∞ and

μrμ (ρ) → 0 a.e.. Hence for f˜ ∈ C0∞ (⺢ \ {0}; X) we obtain μ(Rμ f )∼ → f˜ and in L1 (⺢; X), and so μRμ f → f and μtRμ f → tf in L∞ (⺢; X); μ(tRμ f )∼ → tf this then implies μRμ f → f in Lp (⺢; X). Since the set of all f ∈ Lp (⺢; X) with f˜ ∈ C0∞ (⺢ \ {0}; X) is dense in Lp (⺢; X), by (8.27) we obtain μRμ → I strongly as μ → ∞, μ ∈ Σ(0, ϑη ). In particular, N (Rμ ) = {0} and there is a closed linear operator B in Lp (⺢; X) such that Rμ = (μ + B)−1 , μ ∈ Σ(0, ϑη ), thanks to Theorem 5.8.3 of [180]. (8.27) now shows that the spectral angle φB of B satisfies φB ≤ θb < π since η > 0 has been arbitrary. Since μrμ (ρ) → 0 as μ → 0 for ρ ∈ ⺢ \ {0}, a similar reasoning yields μ(μ + B)−1 → 0 strongly as μ → 0, μ ∈ Σ(0, ϑη ) and so B(μ + B)−1 → I strongly. In particular R(B) is dense in X, hence N (A) = 0, and therefore B belongs to S(X). To complete the construction of B, by the uniqueness of the Fourier transform and the density of functions f with f˜ ∈ C0∞ (⺢\{0}; X) in Lp (⺢; X), it is clear that (8.24) determines B uniquely. As 0 < μ → ∞, we have μ(1 + μˆb(iρ))−1 → ˆb(iρ)−1 a.e., hence as above μB(μ + B)−1 f →μ→∞ Bf for each f˜ ∈ C0∞ (⺢ \ {0}; X), i.e. B satisfies (8.24). iρτ ˜ To verify (i) it is sufficient to observe the relation (T f (ρ), ρ, τ ∈ ⺢, τ f )(ρ) = e 1 valid for each f ∈ L (⺢; X). Next we show (ii), i.e. the causality of B. For this purpose observe that there are functions bμ ∈ L1loc (⺢+ ) such that ˆbμ (λ) =

ˆb(λ) , 1 + μˆb(λ)

Re λ > 0, μ > 0.

For f ∈ C0∞ (⺢+ ; X) we therefore obtain bμ ∗ f (λ) = bμ (λ)fˆ(λ) =

ˆb(λ) fˆ(λ), 1 + μˆb(λ)

Re λ > 0.

Passing to the horizontal limit λ = ε + iρ → iρ this yields ∗ f (iρ) = rμ (ρ)fˆ(iρ), bμ

for a.a. ρ ∈ ⺢,


225

and so we have the representation Rμ f = bμ ∗ f for all f ∈ C0∞ (⺢+ ; X), by uniqueness of the Fourier transform. This implies causality. In the next step we prove |B is | ≤ Kp (1 + |s|)e|s|θb ,

s ∈ ⺢,

(8.29)

i.e. B ∈ BIP(Lp (⺢; X)) and θB ≤ θb . By Proposition 8.1, it is sufficient to prove (8.29) for Bε = (ε + B)(1 + εB)−1 , uniformly for 1 > ε > 0. It is apparent from the preceding parts of this proof that the symbol of Bε is given by ε+z mε (ρ) = (ε + ˆb(iρ)−1 )/(1 + εˆb(iρ)−1 ) = ◦ ϕ(ρ), 1 + εz

ρ = 0,

where ϕ(ρ) = 1/ˆb(iρ). Since ϕ(ρ) is θb -sectorial and the function hε (z) = (ε+z)(1+ εz)−1 preserves any sector Σ(0, φ), φ < π, we have | arg mε (ρ)| ≤ θb , ρ ∈ ⺢ \ {0}, ε ∈ (0, 1). From this it is immediate that the multiplier mε (ρ)is which corresponds to Bεis satisfies |mε (ρ)is | ≤ e|s|| arg mε (ρ)| ≤ e|s|θb ,

ρ ∈ ⺢, ρ = 0,

and ρ

d ϕ(ρ)h ε (ϕ(ρ)) ρϕ (ρ) (mε (ρ)is ) = ismε (ρ)is · · dρ hε (ϕ(ρ)) ϕ(ρ) = ismε (ρ)is ·

(1 − ε2 )ϕ(ρ) · (−iρ · (1 + εϕ(ρ))(ε + ϕ(ρ))

ˆb (iρ) ) ˆb(iρ)

then implies with 1-regularity of b(t) by Lemma 8.1 |ρ

d (mε (ρ)is )| ≤ C|s|e|s|θb , dρ

ρ ∈ ⺢, ρ = 0, s ∈ ⺢.

Theorem 8.2 therefore yields (8.29) for Bε , uniformly for ε ∈ (0, 1), where Kp depends only on p, X, and b, but not on s ∈ ⺢. To prove (iv) observe that the symbol of (μ − B)−1 is given by r−μ (ρ) for μ ∈ ρ(B); therefore μ ∈ ρ(B) implies |μ − 1/ˆb(iρ)| ≥ c > 0 for each ρ ∈ ⺢ \ {0}, i.e. σ(B) ⊃ {1/ˆb(iρ) : ρ ∈ ⺢ \ {0}}. Conversely, if μ ∈ {1/ˆb(iρ) : ρ ∈ ⺢ \ {0}} then

|μ − 1/ˆb(iρ)| ≥ c > 0 for ρ ∈ ⺢ \ {0}. Thus |r−μ (ρ)| ≤ 1/c and also |ρr−μ (ρ)| ≤ C by 1-regularity of b(t), and so Theorem 8.2 yields μ ∈ ρ(B). Finally, we have from (iv) θb ≤ φB ≤ θB ≤ θb , hence θb = φB = θB . 2

Observe that for each component C of ρ(B) we either have (μ − B)−1 Lp (⺢+ ; X) ⊂ Lp (⺢+ ; X) for all μ ∈ C or for none; this follows from the expansion −1

(μ − B)

=

∞

(μ0 − B)−n (μ − μ0 )n

n=0

226


and the connectedness of C. In particular, from the construction of Bεis in Section 8.1 and (ii) of Theorem 8.6 we obtain Bεis Lp (⺢+ ; X) ⊂ Lp (⺢+ ; X), hence B is has the same property since Bεis → B is strongly as ε → 0. Of interest in applications is also the domain D(B) of B. We want to compare D(B) with the fractional power domains D(B0α ), where B0 = −(d2 /dt2 ). It is well known that B0 ∈ S(X), and ∞ √ 1 −1 ((λ + B0 ) f )(t) = √ e− λ(t−s) f (s)ds, t ∈ ⺢, f ∈ Lp (⺢; X); 2 λ −∞ i.e. σ(B0 ) ⊂ [0, ∞) and φB0 = 0. Since the symbol of B0 is m0 (ρ) = ρ2 , by application of Theorem 8.2 it follows as in the proof of Theorem 8.5 that B0 ∈ BIP(Lp (⺢; X)) and θB0 = 0. Define then α/2

H α,p (⺢; X) = D(B0

),

α ∈ ⺢+ .

(8.30)

It is then clear that H 2m,p (⺢; X) = W 2m,p (⺢; X), m ∈ ⺞ and Theorem 8.1 yields H 2mα,p (⺢; X) = [Lp (⺢; X); W 2m,p (⺢; X)]α ,

α ∈ (0, 1), m ∈ ⺞.

α/2

Since the symbol of B0 is given by |ρ|α , we also have the following characterization in terms of Fourier transforms. Let f ∈ Lp (⺢; X); then α˜ f ∈ H α,p (⺢; X) ⇔ there exists fα ∈ Lp (⺢; X) such that f α (ρ) = |ρ| f (ρ),

for ρ ∈ ⺢. Here the Fourier transform is meant in the distributional sense. After these preparations we can now prove Corollary 8.1 Let the assumptions of Theorem 8.6 hold, let B be defined by (8.24), and let α, β ≥ 0. Then (i) limμ→∞ |ˆb(μ)|μα < ∞ implies D(B) → H α,p (⺢; X); (ii) limμ→∞ |ˆb(μ)|μβ > 0 and limμ→0 |ˆb(μ)| > 0 imply H β,p (⺢; X) → D(B). Proof: (i) The goal is to show that mα (ρ) = |ρ|α · ˆb(iρ)(1 + ˆb(iρ))−1 is a multiplier for Lp (⺢; X); this will follow from Theorem 8.2. For |ρ| ≤ 1, we have |mα (ρ)| ≤ |r1 (ρ)| ≤ C, while for |ρ| ≥ 1 by assumption |mα (ρ)| ≤ C. The derivative of mα (ρ) results to ρm α (ρ) = (ρˆb (iρ)/ˆb(iρ))mα (ρ)(1 + ˆb(iρ))−1 + αmα (ρ), from which the boundedness of |mα (ρ)| + |ρm α (ρ)| follows by 1-regularity of b and Lemma 8.1(v), and so Theorem 8.2 applies. (ii) Here we have to prove that mβ (ρ) = [ˆb(iρ)(1 + |ρ|β )]−1 is a multiplier for p L (⺢; X). Boundedness of mβ (ρ) near ρ = 0 is equivalent to |ˆb(iρ)| ≥ c > 0 for |ρ| ≤ 1, and near ρ = ∞ to |ˆb(iρ)||ρ|β ≥ c > 0; this then follows by assumption. For the derivative of mβ (ρ) we obtain ρm β (ρ) = −(iρˆb (iρ)/ˆb(iρ))mβ (ρ) − β|ρ|β (1 + |ρ|β )−1 mβ (ρ),

ρ = 0,


227

and so boundedness of ρm β (ρ) follows from 1-regularity of b. Theorem 8.2 yields the assertion. 2

Observe that by a well known Abelian theorem (cf. Widder [339], Thm. V.I.I.), t lim t−α b(s)ds = bα · Γ(α + 1), for some α ≥ 0, t→0

0

implies lim μαˆb(μ) = bα ,

μ→∞

hence we obtain |ˆb(μ)| ∼ μ−α as μ → ∞, in case bα = 0. This observation yields with Corollary 8.1 Corollary 8.2 Let the assumptions of Theorem 8.5 hold, B be defined by (8.24), and assume in addition (i) limμ→0+ |ˆb(μ)| > 0; t (ii) limt→0 t−α 0 b(s)ds = 0, ∞ exists, for some α ≥ 0. Then D(B) = H α,p (⺢; X). 8.5 Maximal Regularity in Lp Let X be ζ-convex and consider the Volterra equation with main part t t s u(t)+ a(t−s)Au(s)ds = a(t−s)( dB(τ )u(s−τ ))ds+f (t), t ∈ J, (8.31) 0

0

0

where a ∈ L1loc (⺢+ ), A closed linear densely defined, B ∈ BVloc (⺢+ ; B(XA , X)) and f ∈ Lp (J; X). By means of Theorems 8.4 and 8.6 we can now prove the following result on maximal regularity in Lp (J; X), where J = [0, T ]. Here H α,p (J; X) = {f|J : f ∈ H α,p (⺢; X)}. Theorem 8.7 Let X belong to the class HT , p ∈ (1, ∞), and assume (i) AωA = A + ωA ∈ BIP(X), for some ωA ≥ 0; (ii) aωa = e−ωa t a(t) is 1-regular and θa -sectorial, for some ωa , θa ≥ 0; (iii) B ∈ BVloc (⺢+ ; B(XA , X)), B(0) = B(0+) = 0; (iv) θa + θA+ωA < π; (v) limμ→∞ |ˆ a(μ)|μα < ∞, for some α ≥ 0. Then (a) for each f = a ∗ g with g ∈ Lp (J; X) there is a unique a.e. strong solution u ∈ H α,p (J; X) ∩ Lp (J; XA ) of (8.31), and there is a constant C(T ) > 0 such that |u|α,p + |Au|p ≤ C(T )|g|p ;

(8.32)

(b) for each f ∈ Lp (J; XA ) there is a unique a.e. strong solution u ∈ Lp (J; XA ) of (8.31) with u − f ∈ H α,p (J; X) and |u|p + |u − f |α,p + |Au|p ≤ C(T )[|f |p + |Af |p ];

(8.33)

228


(c) for each f ∈ Lp (J; X) there is a unique a.e. mild solution u ∈ Lp (J; X) with a ∗ u ∈ Lp (J; XA ) ∩ H α,p (J; X), and |u|p + |a ∗ u|α,p + |Aa ∗ u|p ≤ C(T )|f |p .

(8.34)

Here | · |α,p denotes the norm of H α,p (J; X). Proof: To prove (a), (8.31) will first be reformulated in the following way. Let t v(t) = e−ωt u(t), g(t) = e−ωt f (t), aω (t) = e−ωt a(t), K(t) = 0 e−ωs dB(s)(ω0 + A)−1 , where ω0 > ωA , is fixed and ω > ωa will be chosen later. Then (8.31) is equivalent to (8.35) v + aω ∗ Av = aω ∗ dK ∗ (ω0 + A)v + aω ∗ g. Define an operator A in Lp (J; X) by means of (Av)(t) = (ω0 + A)v(t),

for a.a. t ∈ J,

with D(A) = Lp (J; XA ); then A ∈ BIP(Lp (J; X)) is invertible and θA ≤ θA+ωA . Define b ∈ L1loc (⺢+ ) as the solution of b = ω0 aω ∗ b + aω . Since |ˆ a(λ)| → 0 with Re λ → ∞, by the Paley-Wiener lemma b ∈ L1 (⺢+ ) provided ω has been chosen large enough, and b is again 1-regular and sectorial, in fact, θb ≤ θa + θ(ω) where θ(ω) → 0 as ω → ∞. Define B ∈ BIP(Lp (J; X)) as the restriction of the operator B constructed in Theorem 8.6 to Lp (J; X); this makes sense in virtue of causality. B is invertible and B −1 v = b ∗ v for all v ∈ Lp (J; X). Observe also that the constants C(η) and Kp are uniform for ω ≥ ω1 , where ω1 is sufficiently large. Thus we can rewrite (8.35) in abstract form as Av + Bv = KAv + g

(8.36)

in Z = Lp (J; X). Since A and B obviously commute and with ω2 sufficiently large, θA + θB ≤ θA+ωA + θa + θ(ω) < π,

for ω ≥ ω2 ,

holds, Theorem 8.4 applies; note that Lp (J; X) belongs to HT since X has this property and 1 < p < ∞. Therefore A + B is invertible and L = (A + B)−1 : Z → ZA ∩ ZB has norm independent of ω ≥ max{ω1 , ω2 }. Thus (8.36) can be rewritten as w = KALw + g (8.37) in Z, where w = (A + B)v. From (iii) we obtain |K|B(Z) ≤

T 0

|dK(t)|B(X) =

T 0

e−ωt |dB(t)|B(XA ;X) |(ω0 + A)−1 |B(X;XA ) → 0


229

as ω → ∞, and since |AL|B(Z) is bounded as ω → ∞, there follows |KAL|B(Z) < 1 for ω sufficiently large. Then (8.37) admits a unique solution w ∈ Z for each g ∈ Z, hence (8.36) admits a unique solution v ∈ ZA ∩ ZB for each g ∈ Z, i.e. (8.31) admits a unique solution u ∈ Lp (J; XA ) for each f ∈ Lp (J; X). Moreover, by Corollary 8.1 we have ZB = D(B) → H α,p (J; X), hence the solution u of (8.31) also belongs to H α,p (J; X). (8.32) follows from the closed graph theorem. This proves (a). To prove (b) we reformulate (8.31) with w = u − f as w + a ∗ Aw = a ∗ dB ∗ w + a ∗ (dB ∗ f − Af ), and so (b) follows from (a). To prove (c) we first solve (8.36) with g = f and obtain v ∈ D(B) ∩ Lp (J; XA ). Then uω = Bv + ω0 v ∈ Lp (J; X) and aω ∗ uω = v. This then shows that u(t) = eωt uω (t) is an a.e. mild solution of (8.31) with the desired properties. Further, if u is an a.e. mild solution then a ∗ u is an a.e. strong solution, hence unique by (a), and so Titchmarsh’s theorem implies uniqueness of a.e. mild solutions. 2

8.6 Strong Lp -Stability on the Halfline As another application of the approach via sums of linear operators from Section 8.3 we consider (8.31) on the halfline. Definition 8.2 Equation (8.31) is called strongly Lp -stable, if for each g ∈ Lp (⺢+ , X) there is a unique a.e. strong solution u ∈ Lp (⺢+ , XA )) of (8.31) on ⺢+ with f = a ∗ g. If (8.31) is strongly Lp -stable then by the closed graph theorem there is a constant C > 0 such that the estimate |u|Lp (⺢+ ;XA ) ≤ C|f |Lp (⺢+ ;XA ) ,

f ∈ Lp (⺢+ ; XA ),

holds, i.e. the solution operator f → u is well-defined and bounded in Lp (⺢+ ; XA ). Theorems 8.5 and 8.6 yield the following result which gives sufficient conditions in the parabolic case; necessary conditions will be derived in Section 10.1. Theorem 8.8 Let X belong to the class HT , p ∈ (1, ∞), A a closed linear operator in X with dense domain, a ∈ L1loc (⺢+ ) of subexponential growth, and assume (i) ωA + A ∈ BIP(X), for some ωA ≥ 0; (ii) a(t) is 1-regular, and limλ→0+ |ˆ a(λ)| > 0; (iii) θa := lim|λ|→∞ | arg a ˆ(λ)| < π − θA+ωA ; (iv) σ(A) ∩ {−1/ˆ a(λ) : Re λ > 0} = ∅. Then there is a constant β > 0 such that equation (8.31) is strongly Lp -stable, for each B ∈ BV (⺢+ ; B(XA , X)) with Var B|∞ 0 < β. Proof: The result will be proved by means of Theorem 8.5. Define the operator A in Z = Lp (⺢+ ; X) by (Au)(t) = Au(t),

for a.a. t > 0, u ∈ D(A) = Lp (⺢; XA );

230


then by assumption (i), we have ωA + A ∈ BIP(Z) and θωA +A ≤ θωA +A . Define B0 on Lp (⺢; X) by (B0 u)∼ (ρ) =

1 ˜ f (ρ), a ˆ(iρ)

ρ ∈ ⺢, f˜ ∈ C0∞ (⺢ \ {0}; X),

and denote by B the restriction of B0 to Z. We show by Theorem 8.6 that ω +B0 ∈ BIP(Lp (⺢; X)) for ω sufficiently large. For this purpose we let aω ∈ L1loc (⺢+ ) be defined by a ˆ(λ)/(1 + ωˆ a(λ)), ω (λ) = a and θω = sup{|arg(λ)| : Re λ > 0}. Since by (ii) and Lemma 8.1, for each R > 0 there is ε = ε(R) > 0 such that |ˆ a(λ)| ≥ ε on ⺓+ ∩ B R (0), for any given η > 0 there is ω(η) such that θω ≤ θa + η

for ω ≥ ω(η).

(8.38)

In fact, by definition of θa there is R = R(η) such that | arg a a(λ) + ω|ˆ a(λ)|2 )| ≤ | arg a ˆ(λ)| ≤ θa + η, ω (λ)| = | arg(ˆ

λ ∈ ⺓+ , |λ| ≥ R(η).

Choose ω(η) ≥ (ε(R(η)) sin θa )−1 , in particular ωε > 1; then for λ ∈ ⺓+ ∩ B R (0) we obtain | arg(ˆ a(λ) + ω|ˆ a(λ)|2 )|

≤ | arg(ˆ a(λ) + ωε|ˆ a(λ)|)| 1 ≤ arcsin ≤ θa , if ω ≥ ω(η), ωε

and so (8.38) follows. Thus aω is 1-regular and θω -sectorial, hence Theorem 4.6 yields ω + B0 ∈ BIP(Z) and θω+B0 ≤ θω . Clearly the restriction B of B0 to Z also satisfies ω + B ∈ BIP(Z) and θω+B ≤ θω by causality, and moreover A and B commute. By (iii) the assumptions of Theorem 8.5 are satisfied, and so A + B is closed with domain D(A) ∩ D(B). To prove the invertibility of A + B by means of Theorem 8.5, it remains to show σ(A) ∩ σ(−B) = ∅. This, however, follows from assumption (iv), since σ(A) ⊂ σ(A) and σ(B) = {1/ˆ a(λ) : Re λ > 0}, which is implied by Proposition 8.2 below. For f = a ∗ g, g ∈ Lp (⺢; X), (8.31) is therefore equivalent to Au + Bu = Ku + g where

t

(Ku)(t) = 0

or to

dB(τ )u(t − τ ),

for a.a. t > 0, u ∈ D(A),

w = K(A + B)−1 w + g, |(A+B)−1 |−1 B(Z,ZA )

w = (A + B)u.

If Var B|∞ then |K(A+B)−1 | < 1, hence I −K(A+B)−1 0 < β = is invertible in Z, and the result follows. 2


231

Proposition 8.2 Suppose X belongs to the class HT , p ∈ (1, ∞), and let b ∈ L1loc (⺢+ ) be of subexponential growth, 1-regular and θ-sectorial, for some θ < π. Let B ∈ S(Lp (⺢; X)) be defined by (8.24) in Theorem 8.6 and denote by B the restriction of B to the subspace Lp (⺢+ ; X). Then σ(B) = {1/ˆb(λ) : Re λ > 0}.

(8.39)

Proof: ⊃: Suppose μ ∈ ρ(B), choose any x ∈ X, x = 0, and x∗ ∈ X ∗ such that < x, x∗ > = 1 and consider f (t) = e−t x. Since f ∈ Lp (⺢+ ; X), there is a unique solution u ∈ Lp (⺢+ ; X) of μu − Bu = f ; therfore ϕ(λ) ˆ =< u ˆ(λ), x∗ > is holomorphic on ⺓+ and satisfies ϕ(λ)(μ ˆ − 1/ˆb(λ)) = 1/(1 + λ),

Re λ > 0.

This implies μ = 1/ˆb(λ) for all Re λ > 0, i.e. {1/ˆb(λ) : Re λ > 0} ⊂ σ(B), and so ⊃ of (8.39) follows from the closedness of σ(B). ⊂ : Conversely, suppose dist(μ; {1/ˆb(λ) : Re λ > 0}) > 0; then by the PaleyWiener lemma, there is bμ ∈ L1loc (⺢+ ) such that ˆbμ (λ) =

ˆb(λ) , ˆ μb(λ) − 1

Re λ > 0.

Therefore, u = bμ ∗ f is the unique solution of u = μb ∗ u − b ∗ f . On the other hand, μ ∈ σ(B) by Theorem 8.6, hence for each f ∈ C0∞ (⺢+ ; X) there is a unique solution v ∈ Lp (⺢; X) of μv − Bv = f . As in the proof of Theorem 8.6 then u ˆ(iρ) = bμ (iρ)fˆ(iρ) = v˜(ρ),

ρ ∈ ⺢ \ {0},

hence u = v by uniqueness of the Fourier transform, and so v ∈ Lp (⺢+ ; X). This implies μ ∈ σ(B). 2

It is known that the Mikhlin multiplier theorem (i.e. (8.6) replaced by (8.9)) is valid also for an operator-valued multiplier acting between two Hilbert spaces; see e.g. Bergh and L¨ ofstrom [22], Theorem 6.1.6. By means of this result, Theorem 8.8 can be extended considerably, in case X and Y are Hilbert spaces. Here we prove only one such result for nonscalar equations; cp. Section 7.2. Theorem 8.9 Let X, Y be Hilbert spaces with continuous and dense embedding d

Y → X, and let A ∈ L1loc (⺢+ ; B(Y, X)) be of subexponential growth. Assume (7.1) is weakly parabolic and A(t) is weakly 1-regular, let S(t) denote the weak resolvent of (7.1), and consider the solution map T defined by (T f )(t) =

d dt

t 0

S(t − τ )f (τ )dτ,

t ≥ 0, f ∈ W 1,1 (⺢+ ; X).

(8.40)

232


Then T ∈ B(Lp (⺢+ ; X)), 1 < p < ∞. If in addition (7.1) is parabolic and A(t) is 1-regular, then T ∈ B(Lp (⺢+ ; Y )). −1 ˆ Proof: Let M (λ) = (I − A(λ)) , Re λ > 0; then

(T f )∧ (λ) = M (λ)fˆ(λ),

Re λ > 0, f ∈ W 1,1 (⺢+ ; X).

By weak parabolicity of (7.1) and weak 1-regularity of A(t), we obtain as in the proof of Theorem 7.2 |M (λ)|B(X) + |λM (λ)|B(X) ≤ C,

Re λ > 0.

Hence the Mikhlin multiplier theorem for operator-valued multipliers on Hilbert spaces implies that there is a constant Cp > 0 such that |e−εt T f |p ≤ Cp |f e−εt |p ,

f ∈ W 1,1 (⺢+ ; X), ε > 0,

with ε → 0 this implies T ∈ B(Lp (⺢+ ; X)). For the second assertion replace X by Y. 2

8.7 Comments a) The theory of complex powers of linear operators was developed independently by Balakrishnan, Kato, and Krasnosel’skii and Sobolevskii. For general expositions of this theory, we refer e.g. to the monographs of Krein [200], Tanabe [319], Triebel [328], and Yosida [347]. Komatsu [199] seems to be the most general treatment which is available presently; there neither D(A) nor R(A) need to be dense in X and N (A) is allowed to be nontrivial. b) The class BIP(X) has been introduced in Pr¨ uss and Sohr [283]. So far no nontrivial characterization of this class seems to be known, even in the Hilbert space case. On the other hand, some permanence properties have been proved; we only mention the following: (i) A ∈ BIP(X) ⇔ A−1 ∈ BIP(X); (ii) A ∈ BIP(X), T ∈ B(X, Z) invertible ⇒ T AT −1 ∈ BIP(Z); (iii) A ∈ BIP(X), 0 < α < π/θA ⇒ Aα ∈ BIP(X). (iv) A ∈ BIP(X), c > 0 ⇒ cA ∈ BIP(X); (v) A ∈ BIP(X), ε > 0 ⇒ ε + A ∈ BIP(X). Here (i)∼(iv) are more or less simple consequences of the definition of Az , and a proof of (v) can be found in Pr¨ uss and Sohr [283]. c) The paper of Baillon and Clément [16] contains examples of sectorial operators in a Hilbert space which do not belong to the class BIP(X). On the other hand, in the Hilbert space case large classes of operators are known which do belong to BIP(X); we mention the following. (i) A normal and sectorial ⇒ A ∈ BIP(X) , θA = φA = supλ∈σ(A) | arg λ|; (ii) A generator of a bounded strongly continuous group ⇒ ±A ∈ BIP(X) and θA = φA = π/2;


233

(iii) A m-accretive ⇒ A ∈ BIP(X), φA ≤ θA ≤ π/2; (iv) A regularly m-accretive with angle ϕA , i.e. A m-accretive and (tan ϕA )|Im(Ax, x)| ≤ Re (Ax, x)

for all x ∈ D(A)

⇒ A ∈ BIP(X) , φA ≤ θA ≤ π/2 − ϕA . Here (i) and (ii) are implied by the functional calculus for normal operators in Hilbert space -in case (ii), A is similar to a skew-adjoint operator-, while (iii) and (iv) are based on the functional calculus for contractions in Hilbert spaces, the H ∞ -calculus; see Foias and Nagy [118]. d) Let Ω ⊂ ⺓ be open. A closed linear densely defined operator A in X is said to have an H ∞ (Ω)-functional calculus if σ(A) ⊂ Ω, and there is a continuous algebra homomorphism f → f (A) from H ∞ (Ω) to B(X), such that (λ − A)−1 = fλ (A) for each λ ∈ Ω, where fλ (z) = (λ − z)−1 . Obviously, if A is sectorial and admits an H ∞ (Σ(0, θ))-functional calculus, then A ∈ BIP(X) and φA ≤ θA ≤ θ; however, the converse of this is in general not true. A admits a H ∞ (Σ(0, θ))-functional calculus, θ < π/2, if and only if −A generates a bounded analytic semigroup in X and ∞ iφ | < Ae−Ate x, x∗ > |dt < ∞, for all x ∈ X, x∗ ∈ X ∗ , |φ| < θ, 0

is satisfied. The latter was recently proved in Boyadzhiev and deLaubenfels [34]. For other results concerning H ∞ -calculus see Boyadzhiev and deLaubenfels [33], Duong [104], [105], McIntosh [238], McIntosh and Yagi [239], and Yagi [343]. e) A Banach space X is said to have the unconditional martingale difference property (UMD) if for each p ∈ (1, ∞) there is a constant Cp such that for any martingale (fn )n≥0 ⊂ Lp (Ω, Σ, μ; X) and any choice of signs (εn )n≥0 ⊂ {−1, 1} and any N ∈ ⺞ the following estimate holds. |f0 +

N

εn (fn − fn−1 )|Lp (Ω,Σ,μ;X) ≤ Cp |fN |Lp (Ω,Σ,μ;X)

n=1

for the notion of vector-valued martingales see Diestel and Uhl [99]. The class of UMD-spaces was introduced by Burkholder who also established the equivalence to ζ-convexity; Burkholder and McConnell proved the implication UMD⇒ HT , and its converse is due to Bourgain [31]. For these results and others see the survey article by Burkholder [35]. f) It is known that a ζ-convex Banach space is superreflexive, and therefore admits a uniformly convex, uniformly smooth equivalent norm. On the other hand, Bourgain [31] contains an example of a superreflexive Banach lattice which is not ζ-convex.

234


Every Hilbert space is ζ-convex, and if (Ω, Σ; μ) is a measure space, p ∈ (1, ∞) then Lp (Ω, Σ, μ; X) is ζ-convex if X has this property. Closed subspaces, quotients, duals, finite products of ζ-convex spaces are again ζ-convex. If X, Y are ζ-convex then the complex interpolation spaces [X, Y ]α as well as the real interpolation spaces (X, Y )α,q are again ζ-convex, provided 1 < q < ∞; cp. Burkholder [35]. g) For a classical proof of the Marcienkiewicz multiplier theorem for Lp (⺢) we refer to Stein [316]. Its vector-valued version, Theorem 8.2, is due to Bourgain [32]. McConnell [237] and Zimmermann [349] contain extensions of this result to the n-dimensional case, i.e. for Lp (⺢n ; X). h) The Marcienkiewicz multiplier theorem can also be used as a starting point to prove that the Laplacian − with Dirichlet or Neumann or Robin boundary conditions belongs to BIP(Lp (Ω)), where Ω ⊂ ⺢n is open, ∂Ω sufficiently smooth, and p ∈ (1, ∞). See Fujiwara [125], Seeley [299] for much more general elliptic uss and Sohr [284] for second differential operators with C ∞ - coefficients, and Pr¨ order operators with C α -coefficients. If positivity is present, i.e. if second order operators in divergence form are considered, the transference principle of Coifman and Weiss [56] can be used; see also Clément and Pr¨ uss [54]. This approach has been carried out by Duong [104]. The Stokes operator introduced in Section 5.6 is also known to have bounded imaginary powers in Lp (Ω; ⺢n ); see e.g. Giga [130] and Giga and Sohr [131]. i) In the situation of Theorem 8.3, Sobolevskii [310] calls the pair (A, B) a coercive pair if A + tB with domain D(A) ∩ D(B) is closed for all t > 0 and there is a constant M > 0 such that |Ax| + t|Bx| ≤ M |Ax + tBx|,

for all x ∈ D(A) ∩ D(B), t > 0.

He then obtains the so-called mixed derivative theorem, i.e. there is a constant C > 0 such that |Aβ B 1−β x| ≤ C|(A + B)x|,

x ∈ D(A) ∩ D(B)

for each β ∈ (0, 1). This result is very useful in particular for nonlinear problems, since e.g. in Theorem 8.7 one obtains the additional regularity u ∈ H αβ,p (J; X1−β ) for the solution of 8.31. j) The results of Section 8.4 generalize and improve Theorem 1 in Pr¨ uss [278] where a ∈ BV (⺢+ ) was assumed. Theorems 8.7 and 8.8 extend Theorem 6 of Pr¨ uss and Sohr [283]; see also Clément and Pr¨ uss [54], where applications to heat flow with memory are given. For applications of results like Theorems 8.7 and 8.8 to nonlinear problems see Clément and Pr¨ uss [55], Giga and Sohr [132], Orlov [265] and Pr¨ uss [279]. k) In Sections 8.5 and 8.6 one may apply of course the Da Prato Grisvard result, Theorem 8.3, instead of the Dore-Venni theorem, Theorem 8.4, but then without


235

further restrictions on the inhomogeneity f no BV0 -perturbations can be included. For the case B ≡ 0 this approach has been worked out in the paper Clément and Da Prato [52] for the whole line case where the kernel is integrable and completely positive. For an interesting application of Theorem 8.3 to equations of the form u + Aa ∗ u + Bb ∗ u = f see Pugliese [285].

9

Viscoelasticity and Electrodynamics with Memory

The discussion of problems in linear viscoelasticity leading to linear abstract Volterra equations which was begun in Section 5 is continued here. Models for viscoelastic beams and plates are introduced and their well-posedness is studied by means of the results on Volterra equations of scalar type from Chapter I but also by those on equations of nonscalar type from this chapter. In Sections 9.3 and 9.4 two approaches to general linear thermoviscoelasticity based on the results of Sections 6, 7, and 8 are carried through. Under physically reasonable assumptions these yield well-posedness in the variational setting, but also in the strong if the material in question is almost separable. In Sections 9.5 and 9.6 memory effects in isotropic linear electrodynamics are discussed and via the perturbation method well-posedness of the whole space problem as well as of a transmission problem are proved. 9.1 Viscoelastic Beams Consider a beam [0, l] × Ω of length l and uniform cross section Ω ⊂ ⺢2 made of a homogeneous isotropic viscoelastic material. W.l.o.g. we assume Ω z dy dz = y dy dz = 0, i.e. (0, 0) is the center of Ω. Let the bending take place only in Ω the (x, z)-plane; then the stress components σ12 and σ23 do not depend on y. For thin beams, i.e. diam Ω = δ 0 is called the shear correction coefficient. The factor κ was introduced to account for the fact that, in general, the shearing strain is not uniform over a cross section of the beam. Balance of momentum then yields the following equations for u1 and u3 . ρ0 u ¨1 = de ∗ u˙ 1xx + κda ∗ (u˙ 1zz + u˙ 3xz ) + ρ0 b1 ,

(9.3)

ρ0 u ¨3 = κda ∗ (u˙ 1zx + u˙ 3xx ) + ρ0 b3 ;

(9.4)


236

9. Viscoelasticity and Electrodynamics

237

observe that for reasons of consistency one should have b2 ≡ 0. If the beam is clamped at one end, say at x = 0, free at the other end, the boundary conditions become u1 (t, 0, z) = u3 (t, 0, z) = 0, (9.5) σ11 (t, l, z) = g1 (t, z),

σ13 (t, l, z) = g3 (t, z),

(9.6)

where g denotes the force acting on the end of the beam; for consistency we assume g2 ≡ 0. With (9.1) and (9.2) boundary conditions (9.6) become de ∗ u˙ 1x (t, l, z) = g1 (t, z),

κda ∗ (u˙ 1z + u˙ 3x )(t, l, z) = g3 (t, z).

(9.7)

At the lateral boundary [0, l] × ∂Ω we obtain σ13 (t, x, y, z)n3 (y, z) = h1 (t, x, y, z),

(9.8)

where h is the force acting on the surface of the beam, and n = (n2 , n3 ) denotes the outer normal of Ω. Again for consistency reasons we have to assume h2 ≡ h3 ≡ 0. With (9.2) this yields the boundary condition n3 κda ∗ (u˙ 1z + u˙ 3x ) = h1

on ⺢ × [0, l] × ∂Ω.

(9.9)

To derive the Timoshenko beam model we introduce new dependent variables, namely 1 w(t, x) = u3 (t, x, z)dy dz, (9.10) α Ω the mean displacement of the beam, φ(t, x) =

1 β

Ω

z u1 (t, x, z)dy dz,

(9.11)

the mean angle of rotation of a cross section of the beam, and 1 ψ(t, x) = u1z (t, x, z)dy dz, α Ω

where

α=

dy dz = mes (Ω), Ω

(9.12)

z 2 dy dz.

β= Ω

Taking the mean value of (9.4) over Ω and the first moment w.r. to z of (9.3) we obtain with an integration by parts and by (9.8) ρ0 w ¨ = κda ∗ (w˙ xx + ψ˙ x ) + fs ˙ + fb , ρ0 φ¨ = de ∗ φ˙ xx − (ακ/β)da ∗ (w˙ x + ψ) where fs =

ρ0 α

Ω

b3 dy dz,

fb =

ρ0 β

Ω

zb1 dy dz +

1 β

zh1 dω. ∂Ω

(9.13)

238


Boundary conditions (9.5) and (9.6) yield t∈⺢

(9.14)

˙ κda ∗ (w˙ x + ψ)(t, l) = gs (t),

(9.15)

w(t, 0) = φ(t, 0) = 0, and de ∗ φ˙ x (t, l) = gb (t), where gb (t) =

1 β

Ω

zg1 (t, z)dy dz

,

gs (t) =

1 α

Ω

g3 (t, z)dy dz,

are the bending moment resp. shear force applied at the free end x = l of the beam. To obtain a complete set of equations, a relation between φ and ψ has to be added. If in addition Ω z 3 dy dz = 0, e.g. if Ω is symmetric w.r. to the (x, y)-plane, Taylor expansion of u1 yields ψ = φ + O(δ 2 ), i.e. for thin beams the approximation ψ ≈ φ is reasonable. The resulting model for thin beams is called viscoelastic Timoshenko beam, in honour of the pioneering work of Timoshenko in this field. ρ0 w ¨ = κda ∗ (w˙ xx + φ˙ x ) + fs ˙ + fb , ρ0 φ¨ = de ∗ φ˙ xx − κγda ∗ (w˙ x + φ)

(9.16)

where γ = α/β, and the boundary conditions in the homogeneous case become w = 0, φx = 0,

φ = 0 at x = 0

wx + φ = 0

at x = l.

(9.17)

Other boundary conditions are possible, we mention here only the supported end w = 0,

φx = 0

and the guided end φ = 0,

φ + wx = 0.

Of interest is also the limiting case κ → ∞, which gives the so called (viscoelastic) Rayleigh beam. As κ → ∞, (9.16) formally yields φ = −wx and elimination of the terms containing κ leads to γρ0 w ¨ − ρ0 w ¨xx + de ∗ w˙ xxxx = f

(9.18)

with f = γfs + fbx . The boundary conditions in the homogeneous case then become for the clamped end w = 0,

wx = 0,


239

for the supported end w = 0,

wxx = 0,

for the guided end ¨x + de ∗ w˙ xxx = 0, wx = 0, −ρ0 w

(i.e. wxxx = 0),

and for the free end wxx = 0,

−ρ0 w ¨x + de ∗ w˙ xxx = 0.

Since γ is of the order δ −2 , in many papers the term ρ0 w ¨xx in (9.18) is neglected; this leads to the viscoelastic Euler-Bernoulli beam ¨ + de ∗ w˙ xxxx = f. γρ0 w

(9.19)

In this situation the boundary conditions for the clamped, supported and guided end are the same as before, while for the free end they become wxx = 0,

wxxx = 0.

To begin the discussion of the well-posedness of the three beam models described above, it is clear that the Euler-Bernoulli beam with any of the boundary conditions at either end considered here leads to Volterra equations of scalar type of the form (1.1) in X = L2 [0, l] and with a selfadjoint operator A which is negative semidefinite. Thus results of Chapter I are applicable, and keeping in mind that the tensile modulus de(t) can be expected to be at least of positive type, well-posedness follows e.g. from Corollary 1.2. Moreover, the discussion of the one-dimensional problems in Section 5.4 apply for the Euler-Bernoulli beam as well. The situation is a little different for the Rayleigh beam. Excluding free ends for the moment, one may proceed as follows. (9.18) can be written as (w.l.o.g. ρ0 = 1) (γ + B)w = e ∗ Aw + g (9.20) in X = L2 [0, l], where A = −(d/dx)4 with D(A) ⊂ W 4,2 [0, l] including the involved boundary conditions, and B = −(d/dx)2 with D(A) ⊂ D(B) ⊂ W 2,2 [0, l] such that B is positive semidefinite; A is negative semidefinite as before. With v = (γ + B)1/2 w, (9.20) can be rewritten as v = e ∗ AB v + h,

(9.21)

where AB = (γ + B)−1/2 A(γ + B)−1/2 is again negative semidefinite. In this way (9.18) has been reduced to a scalar Volterra equation in L2 [0, l]; note that w then lives in its ‘natural space’ W 1,2 [0, l]. This approach does not work if one end of the beam is free since then one of the corresponding boundary conditions contains the kernel de. In this case one may still use a variational approach as in Section 6.7, where H = W 1,2 [0, l], V = W 2,2 [0, l], modulo low order boundary conditions. The resulting resolvent will be subject to the energy inequality |S(t)|B(H) ≤ 1.

240


Turning to the viscoelastic Timoshenko beam, we observe first that (9.16) is of the form (1.1) in X = L2 [0, l] × L2 [0, l], provided the material is synchronous; cp. Section 5.4 (iii). The operator A is defined by the matrix κρ−1 ∂x2 , κρ−1 ∂x 0 0 A= , −1 2 −κγρ−1 0 ∂x , βe ∂x − κγρ0 where βe = 9β/[ρ0 (3β + 1)], β from Section 5.4 (iii), and the domain of A, D(A) ⊂ W 2,2 [0, l] × W 2,2 [0, l] contains the boundary conditions in question. With any of the boundary conditions discussed above at either end, A is negative semidefinite, and so results of Chapter I are applicable. In case the material is not synchronous then (9.16) will not be of scalar type. In this situation we may apply the results of Section 6, in particular Theorems 6.4. and 6.5. In fact, with a(t)∂x2 , κρ−1 a(t)∂x κρ−1 0 0 A(t) = −1 2 −1 −κγρ−1 0 a(t)∂x , e(t)ρ0 ∂x − κγρ0 a(t) in X = L2 [0, l] × L2 [0, l], Y ≡ D(A(t)) ⊂ W 2,2 [0, l] × W 2,2 [0, l] incorporating the boundary conditions in question, (9.16) can be rewritten as (6.1). It is easy to check that dA(t) is of positive type in X, dA(λ) is negative semidefinite for each λ > 0, provided only da and de are of positive type. Therefore Theorem 6.4 yields a weak resolvent for (9.16) in X, which is even an a-regular resolvent by Corollary 6.6 if in addition there is k ∈ BVloc (⺢+ ) with k(0+) > 0 such that b = dk ∗ a; such materials will be called almost synchronous. We may apply Corollary 6.7 instead if both ends are either guided or supported to obtain a resolvent for arbitrary isotropic and homogeneous materials. For example, if the end at x = 0 is supported and that at x = l is guided we have Y = {(w, φ) ∈ W 2,2 ([0, l]; ⺢2 ) : w(0) = φx (0) = φ(l) = wx (l) = 0}, and it is easy to check that Z = {(w, φ) ∈ W 4,2 ([0, l]; ⺢2 ) ∩ Y : wxx (0) = φxxx (0) = φxx (l) = wxxx (l) = 0} satisfies the assumptions of Corollary 6.7. On the other hand, if a free or a clamped end are involved it is not clear whether such a space Z exists, and so in the not almost synchronous case no more regularity of the weak resolvent can be obtained by the results of Section 6.6. But still the variational approach of Section 6.7 can be applied, which gives some more regularity, at least. We leave the developement of this to the interested reader. 9.2 Viscoelastic Plates Similarly to the derivation in Section 9.1, models for thin viscoelastic plates may be obtained. We shall not carry this through here, but simply state the resulting models, and refer to Lagnese and Lions [204] for more details.


241

Let the plate be represented by a bounded domain Ω ⊂ ⺢2 located in the (x, y)-plane. In the Midlin-Timoshenko plate model there are three dependent variables, the mean normal displacement w, and the mean angles of rotation ψ, φ of the cross sections x =const, y = const. Assuming again that the material is homogeneous and isotropic with mass density ρ0 and material functions a(t) and b(t), the equations are ρ0 w ¨ = κda ∗ [w˙ xx + w˙ yy + ψ˙ x + φ˙ y ] + fw ρ0 ψ¨ = da ∗ [ψ˙ xx + ψ˙ yy ] + dc ∗ [ψ˙ xx + φ˙ xy ] − γκda ∗ [wx + ψ] + fψ (9.22) ρ0 φ¨ = da ∗ [φ˙ xx + φ˙ yy] + dc ∗ [φ˙ yy + ψ˙ xy ] − γκda ∗ [wy + φ] + fφ for t ∈ ⺢, (x, y) ∈ Ω ⊂ ⺢2 . Here κ denotes the shear correction coefficient, γ = 6/δ 2 , δ > 0 the thickness of the plate, and the modulus c(t) is defined by the relation db(λ) 9da(λ) dc(λ) = , λ > 0. (9.23) 3db(λ) + 4da(λ) Observe that as in Section 5.4 (iii), c(t) exists and is a Bernstein function if a(t) and b(t) are such; in the synchronous case b(t) = βa(t) we have c(t) = [9β/(3β +4)]a(t), i.e then (9.22) involves only one kernel, hence is of scalar type. The boundary conditions on the part Γc ⊂ ∂Ω where the plate is clamped are w = ψ = φ = 0,

t ∈ ⺢, (x, y) ∈ Γc ,

(9.24)

on the free part Γf ⊂ ∂Ω we have ˙ = gw κda ∗ [∂ w/∂n ˙ + nx ψ˙ + ny φ] da ∗ [nx (ψ˙ x − φ˙ y ) + ny (ψ˙ y + φ˙ x )] + dc ∗ [nx (ψ˙ x + φ˙ y )] = gψ da ∗ [ny (φ˙ y − ψ˙ x ) + nx (ψ˙ y + φ˙ x )] + dc ∗ [ny (ψ˙ x + φ˙ y )] = gφ ,

(9.25)

where n = (nx , ny ) denotes the outer normal of Ω at (x, y) ∈ ∂Ω. Another boundary condition of interest on Γs ⊂ ∂Ω, the simply supported part, is w = ˙ ˙ ˙ ˙ ˙ ˙ da ∗ [nx (ψx − φy ) + ny (ψy + φx )] + dc ∗ [nx (ψx + φy )] = da ∗ [ny (φ˙ y − ψ˙ x ) + nx (ψ˙ y + φ˙ x )] + dc ∗ [ny (ψ˙ x + φ˙ y )] =

0 0

(9.26)

0.

In the limiting case κ → ∞ one obtains the viscoelastic Kirchhoff plate model, i.e. ψ = −wx , φ = −wy , and γρ0 w ¨ − ρ0 Δw ¨ + (da + dc) ∗ Δ2 w˙ = f,

t ∈ ⺢, (x, y) ∈ Ω.

The boundary conditions become on the clamped part ⺢ × Γc w = ∂w/∂n = 0,

(9.27)

242


on the supported part ⺢ × Γs w (da + dc) ∗ Δw˙ − 2da ∗

(n2y w˙ xx

+

n2x w˙ yy

= 0

− 2nx ny w˙ xy ) = 0,

and on the free part ⺢ × Γf (da + dc) ∗ Δw˙ − 2da ∗ (n2y w˙ xx + n2x w˙ yy − 2nx ny w˙ xy ) = g −ρ0 ∂ w/∂n ¨ + (da + dc) ∗ ∂(Δw)/∂n ˙ + dc ∗ (w˙ xxy − w˙ yyx )(nx − ny ) = h. Observe that besides the dimension of the underlying domain, there is a fundamental difference between the beam models discussed in Section 9.1 and the corresponding plate models. Namely both kernels da, dc characteristic for plates not only enter the equations but also the boundary conditions mentioned here, except for an entirely clamped plate. As a consequence, for nonsynchronous materials so far only the variational approach of Sections 6.7 and 7.3 can be used to establish well-posedness and existence of the resolvent, unless the plate is entirely clamped. In the synchronous case the Midlin-Timoshenko plate is of scalar type, and the Kirchhoff plate is so if the boundary ∂Ω contains no free part, i.e. Γf = ∅. Since βe = 9β/(3β + 4) ≤ 3 it is not difficult to show that the corresponding operators A of these models are selfadjoint and negative semidefinite in X = L2 (Ω)3 for (9.22) and X = L2 (Ω) for (9.27). Therefore the results of Chapter I apply again. If the plate is entirely clamped we obtain similarly to the case of beams a weak resolvent by means of Theorem 6.4, which is also an a-regular resolvent if the material is almost synchronous, thanks to Corollary 6.6. In the general case the variational approach can be used if both, da(t) and dc(t) are of positive type, i.e. e.g. if a(t) and b(t) are Bernstein functions. However, we shall not do this here since Section 9.4 is entirely devoted to the variational approach to the more complicated problem of three-dimensional thermoviscoelasticity, and we want to avoid too many repetitions. 9.3 Thermoviscoelasticity: Strong Approach Consider a 3-dimensional body which is represented by a bounded open domain Ω ⊂ ⺢3 with smooth boundary ∂Ω, and let ρ0 ∈ C(Ω), ρ0 > 0, be its density of mass. According to Section 5.3, linear thermoviscoelasticity of this body is governed by the equations ρ0 (x)v(t, ˙ x) =

div S(t, x) + ρ0 (x)g(t, x)

ε(t, ˙ x) = − div q(t, x) + r(t, x) ∞ ∞ ˙ ˙ − τ, x) (9.28) S(t, x) = dA(τ, x)E (t − τ, x) − dB(τ, x)Θ(t 0 0 ∞ ∞ ˙ − τ, x) ε(t, x) = dm(τ, x)Θ(t − τ, x) + dB(τ, x) : E(t 0 0 ∞ q(t, x) = − dC(τ, x)∇Θ(t − τ, x), 0


243

for t ∈ ⺢, x ∈ Ω, where v, ε, S, q, θ, g, r have the same meaning as in Section 5. For the formulation of the boundary conditions, let Γd , Γs , Γb , Γf ⊂ ∂Ω be closed, ◦

◦

◦

◦

such that Γd ∩ Γs = ∅, Γb ∩ Γf = ∅, Γd ∪ Γs = Γb ∪ Γf = ∂Ω, and let n(x) denote the outer normal of Ω at x ∈ ∂Ω. Then v(t, x) = vd (t, x), S(t, x)n(x)

= gs (t, x),

◦

t ∈ ⺢, x ∈Γd , ◦

t ∈ ⺢, x ∈Γs ,

(9.29)

◦

Θ(t, x)

= Θb (t, x),

t ∈ ⺢, x ∈Γb ,

−q(t, x) · n(x)

= qf (t, x),

t ∈ ⺢, x ∈Γf ,

◦

are the natural boundary conditions for (9.28). In the sequel we shall always assume that these are homogeneous, i.e. vd ≡ gs ≡ Θb ≡ qf ≡ 0; as usual, this can be achieved by suitable transformations of the variables v, ε, S, q, Θ, g and the forcing functions g and r, if the inhomogeneities in (9.29) are sufficiently smooth. To obtain a reformulation of (9.28) and (9.29) as an abstract Volterra equation of the form (6.1), i.e. t u(t) = f (t) + A(t − τ )u(τ )dτ, t ≥ 0, (9.30) 0

d

where A ∈ BVloc (⺢+ ; B(Y, X)), Y → X, the spaces X, Y and the variable u must be chosen carefully. We intend to work in an L2 -setting, and choose X = L2 (Ω; ⺢3 ) × L2 (Ω; Sym{3}) × L2 (Ω) × L2 (Ω; ⺢3 ); the variable u will be ˙ q), u = (bm ∗ v, ˙ S, be ∗ Θ, where the kernels bm , be ∈ L1loc (⺢+ ) are still to our disposal. Next we introduce two operators Tm , Te which take care of the differentiations w.r.t. space involved in (9.28) as well as of the boundary conditions (9.29). Te Θ = −∇Θ, Tm v =

1 (∇v + (∇v)T ), 2

Θ ∈ D(Te ) = WΓ1,2 (Ω), b

v ∈ D(Tm ) = WΓ1,2 (Ω; ⺢3 ). d

(9.31)

Observe that Te : D(Te ) ⊂ L2 (Ω) → L2 (Ω; ⺢3 ) is linear, closed, densely defined, and the range R(Te ) is closed in L2 (Ω; ⺢3 ) by Poincaré’s inequality. Similarly, Tm : D(Tm ) ⊂ L2 (Ω; ⺢3 ) → L2 (Ω; Sym{3}) is linear, closed, densely defined, with range R(Tm ) closed in L2 (Ω; Sym{3}) by Korn’s second inequality. Also of interest here are the adjoints of Te , Tm ; they are given by Te∗ q = div q,

∗ Tm E = −div E

q ∈ D(Te∗ ) = {q ∈ L2 (Ω; ⺢3 ) : div q ∈ L2 (Ω), q · n = 0 on Γf } (9.32) ∗ 2 2 3 E ∈ D(Tm ) = {E ∈ L (Ω; Sym{3}) : div E ∈ L (Ω; ⺢ ), En = 0 on Γs }.

244


Here the divergence as well as the boundary conditions appearing in D(Te∗ ) and ∗ D(Tm ) have to be understood in the weak sense. With these notations and eliminating ε from (9.28) the initial value problem for (9.28), (9.29) can be reformulated in the following way. Let A0 ∈ L1loc (⺢+ ; L∞ (Ω; B(Sym{3}))), C0 , B1 , B2 ∈ L1loc (⺢+ ; L∞ (Ω; Sym{3}), and l ∈ L1loc (⺢+ ; L∞ (Ω)) be such that the relations A = bm ∗ A0 , C B ∗ be

= be ∗ C0 , be = l ∗ dm, B = 1 ∗ be ∗ B1 = m ∗ bm ∗ B2

are valid. Then (9.28), (9.29) becomes (9.30), where ⎛ ∗ 0 0 −bm (t)ρ−1 0 Tm ⎜ A0 (t)Tm 0 −B 1 (t) A(t) = ⎜ ⎝ −B2 (t) : Tm 0 0 0 0 C0 (t)Te

⎞ 0 ⎟ 0 ⎟, −l(t)Te∗ ⎠ 0

(9.33)

(9.34)

∗ and the space Y is given by Y = D(Tm )×D(Tm )×D(Te )×D(Te∗ ) equipped with its natural norm; obviously, A ∈ L1loc (⺢+ ; B(Y, X)). Relations (9.33) can be fulfilled in fairly general situations, it only prevails restrictions on the behaviour of m near t = 0 and smoothness of the coupling kernel B(t). In fact, suppose m(·, x) is a creep function for each x ∈ Ω with uniformly positive instantaneous heat capacity 1,1 m(0+, x) ≥ m0 > 0, and let B ∈ Wloc (⺢+ ; L∞ (Ω; Sym{3})), B(0) = B(0+) = 0, B˙ ∈ BVloc (⺢+ ; L∞ (Ω; Sym{3})). Then any functions bm , be ∈ L1loc (⺢+ ) such that

b m ∗ km = 1 = be ∗ ke for some km , ke ∈ L1loc (⺢+ ) will do; we have ˙ B2 = l ∗ km ∗ dB˙ A0 = km ∗ dA, C0 = ke ∗ dC, B1 = ke ∗ dB, where l ∈ L1loc (⺢+ ; L∞ (Ω)) is the solution of l ∗ dm = be which exists by m0 > 0. Having reformulated (9.28), (9.29) as a Volterra equation of nonscalar type (9.30), we are now in position to apply the results obtained in Sections 6, 7, and 8. Of course, one cannot hope for well-posedness of (9.30) with A(t) given by (9.34) without further hypotheses on the material functions, in particular on A(t), C(t), and m(t). In the remainder of this subsection we mainly concentrate on the parabolic case and apply some of the results of Section 7 and 8. (i) Let us first consider only viscoelasticity, i.e. the system for bm ∗ v˙ and S, assuming B(t) ≡ 0. Suppose A(t, x) is of the form t A(t, x) = a(t)A1 (x) + a(t − τ )dA2 (τ, x), t ≥ 0, x ∈ Ω, (9.35) 0

where a(t) is a Bernstein Function, A1 ∈ L∞ (Ω; B(Sym{3})) is uniformly positive definite and A2 ∈ BVloc (⺢+ ; L∞ (Ω, B(Sym{3}))), A2 (0, x) = A2 (0+, x) = 0 a.e. in Ω. Materials for which A(t, x) = a(t)A1 (x), a, A1 as above, will be called (viscoelastically) separable, while those which are subject to (9.35) are termed


245

(viscoelastically) almost separable. Define bm ∈ L1loc (⺢+ ) by ˆbm (λ) = a ˆ(λ). bm (t) exists and is completely monotonic; cp. Lemma 4.2. The equation for the purely viscoelastic problem then has the main part bm (t)A1 , where ∗ 0 −ρ−1 0 Tm A1 = (9.36) A1 Tm 0 is skew-adjoint w.r.t. the inner product on X1 = L2 (Ω; ⺢3 ) × L2 (Ω; Sym{3}) defined by ((u1 , u2 ), (v1 , v2 ))X1 = u1 (x) · v1 (x)ρ0 (x)dx + A−1 1 (x)u2 (x) : v2 (x)dx, Ω

Ω

∗ ). Thus A1 generates a unitary and we choose Y1 = D(A1 ) = D(Tm ) × D(Tm group in X1 , and therefore admits bounded imaginary powers which satisfy |Ais 1 | ≤ e|s|π/2 , s ∈ ⺢. Therefore in case A2 ≡ 0 this problem is always well-posed, and it is also for A2 ≡ 0 provided A˙ 2 ∈ BVloc (Ω; L∞ (Ω; B(Sym{3}))), by Theorem 6.1. The problem is parabolic if in addition | arg a ˆ(λ)| ≤ Θa < π holds for Re λ > 0; then without further assumptions on A2 (t, x) we obtain the well-posedness by Theorem 7.6 as well as maximal regularity of type C α and Lp by Theorems 7.4 and 8.7. ˙ and (ii) Similarly one considers the purely thermodynamical problem for be ∗ Θ q in case B(t) ≡ 0. Materials which are subject to

m(t, x) = m(t)σ0 (x),

C(t, x) = c(t)C1 (x)

(9.37)

with m, c Bernstein functions, σ0 ∈ L∞ (Ω), σ0 (x) ≥ σ00 > 0 a.e., and C1 ∈ L∞ (Ω; Sym{3}) uniformly positive definite, will be called (thermodynamically) 1 ˆ separable. Choosing l ∈ L (⺢+ ) such that l(λ) = cˆ(λ)/dm(λ), be = l ∗ dm, as loc

in Section 5.5, the function l(t) is completely monotonic, and the thermodynamical problem becomes of scalar type, with kernel l(t)A2 , where 0 −σ0−1 Te∗ A2 = . C1 Te 0 Similar to (i), A2 is skew-adjoint w.r.t. a suitable inner product on X2 = L2 (Ω) × L2 (Ω; ⺢3 ), and the conclusions are analogous to (i), also for the (thermodynamically) almost separable case t m(t, x) = m(t)σ0 (x) + m(t − τ )dσ1 (τ, x) 0 t C(t, x) = c(t)C1 (x) + c(t − τ )dC2 (τ, x) (9.38) 0

with m, c, σ0 , C1 as above, σ1 ∈ BVloc (⺢+ ; L∞ (Ω)), σ1 (0, x) = σ1 (0+, x) = 0, and C2 ∈ BVloc (⺢+ ; L∞ (Ω; Sym{3})) such that C2 (0, x) = C2 (0+, x) = 0. Observe that the problem is parabolic if | arg(ˆ c(λ)/dm(λ))| ≤ Θl < π for Re λ > 0.

246


(iii) Finally, consider the thermoviscoelastic case B(t) ≡ 0, when the material is almost separable, both viscoelastically and thermodynamically. The coupling involving B(t) will be considered as a perturbation, therefore we must have B2 = l ∗ dB3 , B3 (0) = B3 (0+) = 0. If e.g. B˙ ∈ BVloc (⺢+ ; L∞ (Ω; Sym{3})) this factorization follows, hence the results of Sections 7 and 8 apply to the result that the full thermoviscoelastic problem is well-posed and enjoys maximal regularity of type C α and Lp , provided the problem is parabolic, viscoelastically but also thermodynamically. In the hyperbolic case Theorem 6.1 implies well-posedness if e.g. B¨ ∈ BVloc (⺢+ ; L∞ (Ω; Sym{3})). 9.4 Thermoviscoelasticity: Variational Approach Consider again problem (9.28), (9.29) where vd ≡ 0 ≡ Θb . We want to rewrite it in variational form. For this purpose assume again m(·, x) ∈ CF for each x ∈ Ω, ∗

σ0 (x) = m(0+, x) > 0. Let V = WΓ1,2 (Ω; ⺓3 ) × WΓ1,2 (Ω); then V → H → V , d b 2 3 2 where H = L (Ω; ⺓ ) × L (Ω) is equipped with the inner product ((v1 ; Θ1 ), (v2 , Θ2 ))H = ρ0 (x)v1 (x) · v2 (x)dx + σ0 (x)Θ1 (x)Θ2 (x)dx. d

Ω

d

Ω

We define a bounded sesquilinear form on V by means of β(t, (v1 , Θ1 ), (v2 , Θ2 )) = ∇v1 (x) : A(t, x)∇v2 (x)dx Ω ˙ x)Θ2 (x)dx + ˙ x) : ∇v2 (x)dx + ∇v1 (x) : B(t, Θ1 (x)B(t, Ω Ω + Θ1 (x)m1 (t, x)Θ2 (x)dx + ∇Θ1 · C(t, x)∇Θ2 (x)dx. Ω

(9.39)

Ω

With u = (v, Θ) there follows the variational formulation of (9.28), (9.29) by an integration by parts. t (w, u(t))H + β(t − s, w, u(s))ds = < w, f (t) >, t ≥ 0, (9.40) 0

∗

for each w ∈ V , where f (t) ∈ V contains g(t, x), r(t, x), gs (t, x), qf (t, x) as well as the histories of v(t, x) and Θ(t, x). Since β(t, ·, ·) is bounded, there is a family ∗ of bounded linear operators {A(t)}t≥0 ⊂ B(V, V ) such that β(t, w, u) = − < w, A(t)u >,

for all u, w ∈ V, t ≥ 0.

(9.41)

∗ ˙ C∈ There follows A ∈ BVloc (⺢+ ; B(V, V )), provided m1 ∈ BVloc (⺢+ ; L∞ (Ω)), B, ∞ BVloc (⺢+ ; L (Ω; Sym{3})), and A ∈ BVloc (⺢+ ; L∞ (Ω; B(Sym{3}))). Thus the variational formulation of (9.28), (9.29) becomes t A(t − τ )u(τ )dτ + f (t), t ≥ 0; (9.42) u(t) = 0

∗

in X = V , Y = V .


247

In the case of homogeneous isotropic materials the form β simplifies to β(t, u1 , u2 ) = 2a(t)[(Ev1 : Ev2 ) − ( div v1 , div v2 )/3] + b(t)( div v1 , div v2 ) + c(t)(∇Θ1 , ∇Θ2 ) + m1 (t)(Θ1 , Θ2 ) + (db ∗ dα)(t)[( div v1 , Θ2 ) − (Θ1 , div v2 )],

t>0

where the moduli a(t), b(t), c(t), α(t), and m(t) = σ0 + 1 ∗ m1 are as in Section 5.3. Let us concentrate on the hyperbolic case, and apply the positive type methods presented in Section 6.7. For this purpose consider the inequalities x)q ≥ 0 for all q ∈ ⺓3 , x ∈ Ω, Re λ > 0, Re q · dC(λ,

(9.43)

and 1 (λ, x)ϑ] ≥ 0, ˙ x)ϑ + ϑd ˙ x) : E + ϑdm Re [E : dA(λ, x)E − E : d B(λ, B(λ,

(9.44)

for all symmetric E ∈ ⺓3×3 , ϑ ∈ ⺓, x ∈ Ω, Re λ > 0. Physically (9.43), (9.44) are related to the second law of thermodynamics; cp. Christensen [42]. These inequalities will be termed dissipation inequalities in the sequel, in analogy to those of pure viscoelasticity. In the homogeneous isotropic case (cp. Section 5.3), they reduce to the requirements that dc is of positive type and 1 2 dαIm(tr 1 |ϑ|2 ] ≥ 0, Re [2da(|E| − |trE|2 ) + db|tr E|2 + 2iλdb Eϑ) + dm 3 for all symmetric E ∈ ⺓3×3 , ϑ ∈ ⺓, Re λ > 0. But this in turn is equivalent to da, db, dc, dm1 of positive type, and existence of a constant δ ≤ 1 such that 2 1 (λ), [Im (λdb(λ) dα(λ))] ≤ δRe db(λ) · Re dm

Re λ > 0;

(9.45)

observe that (9.45) holds for classical thermoelasticity given by (5.41). The dissipation inequalities imply u, u) ≤ 0, Re < u, dA(λ)u > = −Re dβ(λ,

u ∈ Y, Re λ > 0,

(9.46)

i.e. the part of dA(λ) in H, dA(λ)| H is dissipative for each Re λ > 0, and the closure in H of dA(λ)|H is even m-dissipative. However, this is not enough for Theorem ∗ 6.4 to apply, neither in H nor in V . One reason for this lies in the weakness of the dissipation inequalities. But even if these are strengthened, Theorem 6.4 still is of ∗ no use here, since in general, dA(λ) will not be dissipative in V and D(dA(λ)| H) will depend on λ. On the other hand, let the dissipation inequalities (9.43) and (9.44) be replaced by the strong dissipation inequalities x)q ≥ γ(Re Re q · dC(λ,

1 )|q|2 , λ+η

Re λ > 0, q ∈ ⺓3 , x ∈ Ω,

(9.47)

248


and 1 (λ, x)ϑ] ˙ x)ϑ + ϑd ˙ x) : E + ϑdm Re [E : dA(λ, x)E + E : d B(λ, B(λ, 1 ≥ γ(Re )|E|2 , λ+η for Re λ > 0, x ∈ Ω, E ∈ ⺓3×3 symmetric, ϑ ∈ ⺓,

(9.48)

where γ > 0 is a constant, and assume one more degree of regularity for the ˙ C, m1 . Then in case η = 0, resp. η > 0, the assumptions of coefficients A, B, ◦

Theorem 6.5, resp. Corollary 6.7, are fulfilled, provided in addition Γd = ∅ and ◦ Γb = ∅ by the inequalities of Korn and Poincaré, and we obtain a resolvent S(t) which enjoys properties (a) ∼ (c) of Theorem 6.5, resp. Corollary 6.7, in particular is leaving H invariant and satisfies |S(t)|H ≤ 1. Observe that in the homogeneous, isotropic case this means σ0 > 0, a0 = b0 = c0 = α0 = 0, a1 (0+), b1 (0+), c1 (0+), m1 (0+), α1 (0+) < ∞, whenever a, b, c, m, α are creep functions (cp. Section 5.3). The strong dissipation inequalities hold if a1 , b1 , c1 , m1 are strongly positive (or merely of positive type if we allow for an exponential shift) and (9.45) holds with δ < 1. Similarly, one can apply Theorem 7.3 in the parabolic case; the translation of conditions (PV1)∼(PV4) into parabolic dissipation inequalities is left to the reader. 9.5 Electrodynamics with Memory As is well known, the macroscopic electrodynamic fields in a medium at rest in the considered inertial system, are governed by Maxwell’s equations Bt + curl E = 0,

div B = 0,

Dt − curl H + J = 0,

div D = ρ,

(9.49)

where E denotes the electric field, H the magnetic field, B the magnetic induction, D the electric induction, J the free current, and ρ the free charge. Clearly (9.49) is underdetermined and constitutive equations have to be supplemented, which specify the electrodynamic properties of the medium in question. For the simplest medium - the vacuum - these relations are given by B = μ0 H,

D = ε0 E,

J = 0,

(9.50)

where ε0 , μ0 > 0 are well known fundamental physical constants which are connected with the speed of light in vacuum c0 by the relation ε0 μ0 = c−2 0 .


249

The next simplest medium is the rigid linear isotropic dieletric which Maxwell defined by B = μH, D = εE, J = 0, (9.51) where ε, μ are material constants (ε ≥ ε0 > 0, μ > 0). Soon it was realized that such simple constitutive relations do not account for the dielectric losses observed, when the medium is placed in a rapidly varying electromagnetic field. This defect cannot be removed by the introduction of a conductivity σ > 0, i.e. replacing the last equation in (9.51) by a relation of the form J = σE (Ohm’s law), i.e B = μH,

D = εE,

J = σE.

(9.52)

More precisely, if one considers periodic fields E(t) = E0 eiωt , D(t) = D0 eiωt , one still observes relations of the form D0 = εÊ0 for each frequency ω > 0, however, εˆ will be complex in general, and depend on ω. This phenomenon is known as electromagnetic dispersion in the physical literature. To account for dispersion in a rigid linear isotropic medium the following constitutive relations have been proposed B = dμ ∗ H,

D = dε ∗ E,

J = dσ ∗ E,

(9.53)

where μ, ε, σ ∈ BV (⺢+ ) are given material functions. Observe that the quantities μ(∞), ε(∞), σ(∞) correspond to the constants appearing in (9.52) in the static (time-dependent) case. Not much seems to be known about general material functions; however, from the discussion in Kapitel IX of Landau and Lifschitz [205] one can guess that ε(t) should be (at least) a bounded creep function and that μ(0+) = μ0 , ε(0+) = ε0 are the fundamental constants of the vacuum introduced above. It is also reasonable to expect that σ(t) is a bounded creep function as well. In the paramagnetic case one has μ(∞) > μ0 , while μ(∞) < μ0 holds for diamagnetic media; therefore in the former case μ(t) can be considered a bounded creep function, while in the latter μ0 − μ(t) should be such. The continuity equation for the charge density ρ which follows from Maxwell’s equation (9.49) reads ρt + div J = 0. (9.54) Convolving (9.54) with dε, (9.53) and (9.49) then imply dε ∗ ρt + dσ ∗ ρ = 0;

(9.55)

considering the history of ρ(t, x) up to t = 0 as known, it is clear that (9.55) determines ρ(t, x) for all t ∈ ⺢, and so in the sequel ρ(t, x) will be considered as known. Assuming the constitutive laws (9.53), Maxwell’s equations can be decoupled as usual. Taking the curl of the first equation of (9.49), with the identity curl curl E = grad div E − ΔE,

250


by div D = ρ and with (9.53) one obtains e.g. for D dε ∗ dμ ∗ Dtt + dμ ∗ dσ ∗ Dt div D

= ΔD − ∇ρ = ρ.

(9.56)

Analogously, applying curl to the third equation of (9.49) leads to the following equation e.g. for B dε ∗ dμ ∗ Btt + dμ ∗ dσ ∗ Bt div B

= ΔB = 0.

(9.57)

Note that the relations (9.53) are invertible; therefore H satisfies again (9.57) while E solves (9.56) with ρ replaced by dν ∗ ρ, where dν ∗ dε = de0 ; e0 (t) as before denotes Heaviside’s function. Thus as in the case of the vacuum, once one of the fields E, D, H, or B is known then Maxwell’s equations and the constitutive equations (9.53) determine the others by pointwise integration resp. convolution. We are now in position to apply the results on scalar Volterra equations obtained in Chapter I, say to (9.57). For simplicity we consider (9.57) on all of ⺢3 . Let X = L2σ (⺢3 ) and denote by A2 the Stokes operator introduced in Section 5.6. With b ∈ BVloc (⺢+ ) defined by −1 ˆb(λ) = (λ2 dε(λ) dσ(λ)) dμ(λ) + λdμ(λ) −1 −1 −1 = (λdμ(λ)) (λdε(λ)) (1 + dσ(λ)/λ dε(λ)) ,

the initial value problem for (9.57) is then equivalent to t b(t − τ )A2 u(τ )dτ, t > 0, u(t) = f (t) +

Re λ > 0,

(9.58)

0

where u(t) means magnetic field or magnetic induction and f is given. Now we can claim the following. t t (i) Assume ε(t) = ε0 + 0 ε1 (s)ds, μ(t) = μ0 ± 0 μ1 (s)ds where ε1 , μ1 ∈ BVloc (⺢+ ), and σ ∈ BVloc (⺢+ ). Then b(t) is of the form b(t) = tc20 + t ∗ r(t), with r ∈ BVloc (⺢+ ), hence Theorem 1.2 applies and shows that (9.58) is well-posed. The resolvent S(t) for (9.58) is of the form S(t) = Co(c0 t) + S1 (t) where Co(τ ) denotes the bounded cosine family generated by the Stokes operator A2 , and S1 (t) is a perturbation which, however, may grow exponentially. (ii) A bounded resolvent can be obtained by means of Corollary 1.2. Suppose ε, μ, σ are bounded creep functions with ε(0+) = ε0 , μ(0+) = μ0 and such that μ1 and σ1 (t) are of positive type, and assume in addition 1 (iρ)/ρ ≥ 0 Re ε1 (iρ) + Im σ

for

ρ > 0.

(9.59)

Then db(t) is of positive type, hence Corollary 1.2 yields boundedness of the resolvent S(t). (iii) Stronger assertions can be obtained by the subordination principle of Section 4. For this assume that ε, μ, σ are bounded Bernstein functions with


251

ε(0+) = ε0 , μ(0+) = μ0 and such that in addition −ε˙1 − σ1 is completely mono t tonic on (0, ∞); note that the latter implies (9.59). Then ε(t) + 0 σ(s)ds is a Bernstein function, hence by Lemma 4.3 there is a Bernstein function k such that ˆ k(λ) = μ ˆ(λ)[ˆ μ(λ)/(ˆ ε(λ) + σ ˆ (λ)/λ)]−1/2 = [ˆ μ(λ)(ˆ ε(λ) + σ ˆ (λ)/λ)]1/2 ,

Re λ > 0.

From this there follows b = c ∗ c for some completely monotonic function c ∈ L1loc (⺢+ ), hence by Theorem 4.3 there is a bounded resolvent S(t) for (9.58) which by Theorem 4.4 has the form S(t) = Co(t/κ)e−αt/κ + S0 (t),

t > 0, √ −1 where κ = k0 = ε0 μ0 = c0 , and α = k∞ + k1 (0+) = k1 (0+), i.e. 1 ε0 1 μ0 α = μ1 (0+) + (ε1 (0+) + σ0 ) . 2 μ0 2 ε0 Thus the (maximum) wave speed is c0 , the speed of light in vacuum, and the resolvent in question decomposes into the resolvent of the vacuum, exponentially damped with attenuation α and a dispersive part S0 (t) which is smooth. Observe that we always have α > 0, unless μ1 (0+) = ε1 (0+) = σ0 = 0, i.e. μ(t) ≡ μ0 , ε(t) ≡ ε0 and σ0 = 0. 9.6 A Transmission Problem for Media with Memory Problems in electrodynamics involving media with memory can in general not be formulated as Volterra equations of scalar type, even if the medium in question is homogeneous and isotropic. We want to discuss briefly a transmission problem for such media in the framework of the perturbation results for nonscalar equations obtained in Section 6.3. Suppose Ω1 ⊂ ⺢3 is a domain with smooth boundary Γ and let Ω2 = ⺢3 \ Ω1 ; Ωi , i = 1, 2 be filled with a homogeneous isotropic medium with material functions 1,1 εi , μi , σi which satisfy the assumptions (i) of Section 9.5, i.e. εi , μi ∈ Wloc (⺢+ ), εi (0+) = ε0 , μi (0+) = μ0 , and ε˙i , μ˙ i , σi ∈ BVloc (⺢+ ). As before, let Bi , Hi , Di , Ei denote the magnetic induction, magnetic field, electrical induction, electric field, respectively, in domain Ωi . Then we have Maxwell’s equations in each domain Ωi , Bit + curl Ei = 0, Dit − curl Hi + Ji = 0,

div Bi = 0, div Di = ρi ,

(9.60)

and the material relations Bi = dμi ∗ Hi ,

Di = dεi ∗ Ei ,

Ji = dσi ∗ Ei ,

(9.61)

for i = 1, 2. At the interface Γ we have to supplement transmission conditions. If n(x) denotes the outer normal at x ∈ ∂Ω1 = Γ of Ω1 these can be stated as n · B1 = n · B2 , n · (D2 − D1 ) = ρs ,

n × E1 = n × E2 n × (H2 − H1 ) = Js

(9.62)

252


on Γ, where ρs denotes the density of surface charges, and Js the density of surface currents; observe that Js · n = 0. The initial value problem for (9.60)∼(9.62) will now be reformulated as a Volterra equation of the form (6.24), i.e. as u˙ = Au + B1 ∗ u + dB0 ∗ u + f,

u(0) = u0 ,

(9.63)

where u = (H1 , E1 , H2 , E2 ) lives in X = L2 (Ω1 ; ⺢3 )2 × L2 (Ω2 ; ⺢3 )2 with inner product (u, v)X = (μ0 u1 · v¯1 + ε0 u2 · v¯2 )dx + (μ0 u3 · v¯3 + ε0 u4 · v¯4 )dx. Ω1

Ω2

We assume that there are no surface currents, i.e. Js ≡ 0. Define an operator A0 in X by −1 −1 −1 A0 u = (−μ−1 0 curl u2 , ε0 curl u1 , −μ0 curl u4 , ε0 curl u3 ),

with D(A0 ) = {u ∈ X : u1 , u2 ∈ H 1,2 (Ω1 ; ⺢3 ), u3 , u4 ∈ H 1,2 (Ω2 ; ⺢3 ), n × (u2 − u4 ) = n × (u1 − u3 ) = 0 on Γ}. It then follows by integration by parts that A∗0 ⊃ −A0 , hence A0 is skew-symmetric, in particular closable. One can also show R(I − A0 ) = X, and therefore the closure A of A0 is skew-adjoint. Thus A generates a unitary C0 -group T (t) in X. In the sequel it is of advantage to invert the material relations for Bi and Di , i.e. Hi = μ−1 Ei = ε−1 (9.64) 0 Bi + νi ∗ Bi , 0 Di + η i ∗ Di where νi , ηi ∈ BVloc (⺢+ ) by the assumptions on εi on μi . Let the Hilbert space Y be defined by Y = {u ∈ X : curl u = (curl ui )4i=1 ∈ X} with inner product (u, v)Y = (u, v)X + (curl u, curl v)X ; then D(A0 ) ⊂ Y and |u|Y ≤ C(|A0 u| + |u|), u ∈ D(A0 ), for some constant C > 0. We can now define B1 ∈ BVloc (⺢+ ; B(Y, X)) by means of B1 (t)u = (−ν1 (t)curl u2 , η1 (t)curl u1 , −ν2 (t)curl u4 , η2 (t)curl u3 ) and B0 ∈ BVloc (⺢+ ; B(Y, X)) by −1 B0 (t)u = (0, ε−1 0 σ1 (t)u2 − (η1 ∗ σ1 )(t)u2 , 0, −ε0 σ2 (t)u4 − (η2 ∗ σ2 )(t)u4 ).

Then (9.60)∼(9.62) is equivalent to (9.63), where u0 = (H1 (0), E1 (0), H2 (0), E2 (0)) and f contains the history of u(t), t ≤ 0. We may apply now the perturbation


253

theorem, Theorem 6.1, in the form of Corollary 6.2, thereby proving the wellposedness of the problem in question for the L2 -setting. The resolvent S(t) for (9.63) obtained this way is of at most exponential growth provided εi , μi , σi have this property, this is the most which can be said, in general. However, if in addition ε˙i , μ˙ i , dσi are of positive type, i = 1, 2, then |S(t)|B(X) ≤ 1. In fact, (9.60)∼(9.62) yield the identity 1 d 2 2 { (ε0 |E1 | + μ0 |H1 | )dx + (ε0 |E2 |2 + μ0 |H2 |2 )dx} 2 dt Ω1 Ω2 =− (ε˙1 ∗ E1 · E1 + μ˙ 1 ∗ H1 · H1 + dσ1 ∗ E1 · E1 dx Ω1 − (ε˙2 ∗ E2 · E2 + μ˙ 2 ∗ H2 · H2 + dσ2 ∗ E2 · E2 )dx, Ω2

hence after integration |u(t)|X ≤ |u0 |X , t ≥ 0. In this situation one can also apply Theorem 6.4 instead of the perturbation result. 9.7 Comments a) For the classical theory of elastic beams and plates we refer e.g. to Landau and Lifschitz [207] and Timoshenko and Woinowsky-Krieger [325]. The recent monograph Lagnese and Lions [204] contains a very readable introduction into classical plate models and their history. For viscoelastic beams and plates see e.g. Bland [24], Fl¨ ugge [117], Lagnese [203], and Lagnese and Lions [204]. b) The derivation of the viscoelastic beam models in Section 9.1 is different from the usual one, where the correspondence principle of viscoelasticity is applied to the classical models of purely elastic beams. We think that the derivation presented here is somewhat more rigorous, and gives more insight where approximations come in. In particular, it seems that neglecting the term ρ0 w ¨xx in (9.18) when passing to the Euler-Bernoulli beam causes the trouble with robustness of the latter; see (c) below. c) There is a recent interest in well-posedness, stability, stabilization and destabilization by boundary feedback laws, and robustness w.r.t. delays in boundary feedback for viscoelastic beams and plates; cf. Desch and Wheeler [94], Desch, Hannsgen, and Wheeler [88], Desch, Hannsgen, Renardy, and Wheeler [87], Grimmer, Lenczewski, and Schappacher [141], Hannsgen and Wheeler [HaWh89], Hannsgen, Renardy, and Wheeler [166], Kim and Renardy [197], Lagnese [203], Lagnese and Lions [204], Leugering [210], [211]. In particular, in [87] it is shown that boundary feedback with delay of the Euler-Bernoulli beam may lead to a loss of well-posedness; on the contrary for the Timoshenko beam this cannot happen [141]. It is not clear whether the Rayleigh beam also has this defect, but from (b) above one might expect that the answer is no. d) Poincaré’s inequality is standard in the modern theory of partial differential equations and can be found in any decent textbook on this subject. For Korn’s

254


inequalities we refer e.g. to Marsden and Hughes [234] or Leis [208]. The approach to thermoviscoelasticity given in Section 9.3 seems to be new. It heavily depends on the almost separability of the material in question, and it is unclear at present whether positive type methods like Theorem 6.4 are can be used here. e) The variational approach to viscoelasticity is originally due to Dafermos [75], [76]; see als Adali [2], Bouc, Geymonat, Jean, and Nayroles [30]. The methods used in these papers are based on perturbation techniques, their results use stronger assumptions than those of Section 9.4. f) For the formulation of Maxwell’s equations (9.49) and the transmission conditions (9.62) see any reasonable textbook on electrodynamics, e.g. Landau and Lifschitz [206], [205], Grant and Phillips [135], Simonyi [305]. g) It seems that already Maxwell knew about the phenomenon of dispersion. Hopkinson [183] was the first to use a constitutive relation of the form D(t) = ε0 E(t) + (ε1 ∗ E)(t),

t ∈ ⺢,

while he kept the relations B = μH and J = σE. By means of these laws he was able to fit his data on the residual charge of a Leyden jar, choosing ε1 (t) of the form ε1 (t) = aj exp(−αj t), t > 0, j

where aj , αj ≥ 0. In Volterra [331] the mathematical study of memory effects in electrodynamics was continued. h) For more recent works on electrodynamics with memory see Toupin and Rivlin [326] and Bloom [25]. The constitutive laws for general time invariant linear media are according to Toupin and Rivlin [326] given by D H

= =

p

aj E (j) +

p

0

0

p

p

0

dj E (j) +

cj B (j) + ϕ1 ∗ E + ϕ2 ∗ B bj B (j) + ψ1 ∗ E + ψ2 ∗ B,

0

where aj , bj , cj , dj are matrices and ϕi , ψi are integrable matrix-valued functions.

Chapter III

Equations on the Line

10

Integrability of Resolvents

The connections between stability properties of Volterra equations and integrability of the corresponding resolvent are discussed in Section 10.1; this discussion motivates the study of integrability of resolvents. For the classes of equations of scalar type introduced in Sections 2, 3, and 4 a complete characterization of integrability of S(t) in terms of spectral conditions is derived. For nonscalar parabolic problems sufficient conditions are presented in a fairly general setting, while for nonscalar hyperbolic problems the analysis is valid in Hilbert spaces only, as counterexamples show. 10.1 Stability on the Halfline Consider the equation of scalar type t u(t) = f (t) + a(t − τ )Au(τ )dτ, 0

t ∈ ⺢+ ,

(10.1)

on the halfline ⺢+ , where a ∈ L1loc (⺢+ ) is Laplace transformable and A is a closed linear operator in X with dense domain D(A). In Section 8.6 we introduced the concept of strong Lp -stability of (10.1) and already obtained sufficient conditions. In general, however, this concept is too strong to be meaningful; below we show that strong Lp -stability of (10.1) for some p ∈ [1, ∞] already implies that (10.1) is parabolic in the sense of Definition 3.1. The following concepts are more appropriate in hyperbolic situations. Definition 10.1 (i) Equation (10.1) is called Lp -stable (R), if for each g ∈ Lp (⺢+ ; X) there is a unique a.e. mild solution u ∈ Lp (⺢+ ; X) of (10.1) with f = a ∗ g. 1,p (ii) Equation (10.1) is called Lp -stable (S) if for each f ∈ Wloc (⺢+ ; X), f˙ ∈ p p L (⺢+ ; X) there is a unique a.e. mild solution u ∈ L (⺢+ ; X) of (10.1). If (10.1) is Lp -stable (R) then for each f ∈ Lp (⺢+ ; XA ) there is a unique a.e. mild solution u ∈ Lp (⺢+ ; X) of (10.1); in fact, u = f + v, where v denotes the a.e. mild solution of v = a ∗ Af + a ∗ Av. Moreover, g ∈ Lp (⺢+ ; XA ) or f, Af ∈ Lp (⺢+ ; XA ) imply u ∈ Lp (⺢+ ; XA ). Similarly, if (10.1) is Lp -stable (S) then f ∈ W 1,p (⺢+ ; XA ) implies u ∈ Lp (⺢+ ; XA ) is an a.e. strong solution of (10.1). To obtain sufficient conditions for Lp -stability in terms of the resolvents S(t) and R(t) associated with (10.1) we have to introduce several notions of integrability of operator families. Definition 10.2 Let {T (t)}t≥0 ⊂ B(X, Z) be a strongly measurable family of operators, i.e. T (·)x is Bochner-measurable in Z, for each x ∈ X. Then T (t) is called (i) strongly integrable, if T (·)x ∈ L1 (⺢+ ; Z) for each x ∈ X; (ii) integrable if there is ϕ ∈ L1 (⺢+ ) such that |T (t)| ≤ ϕ(t) a.e. on ⺢+ ; (iii) uniformly integrable, if T (·) ∈ L1 (⺢+ ; B(X, Z)). J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_10, © Springer Basel 1993

256

10. Integrability of Resolvents

257

Obviously, every uniformly integrable operator family T (t) is integrable, however not conversely, unless T (·) is Bochner-measurable in B(X, Z). Similarly, every integrable family T (t) is also strongly integrable, but the converse is not true in general. Suppose now (10.1) admits an integrable and bounded resolvent S(t) and let 1,p (⺢+ ; X), such that f˙ ∈ Lp (⺢+ ; X), 1 ≤ p ≤ ∞. Then the mild solution f ∈ Wloc of (10.1) is given by the variation of parameters formula t u(t) = S(t)f (0) + S(t − τ )f˙(τ )dτ, t ∈ ⺢+ , (10.2) 0

1,p and therefore we obtain u ∈Lp (⺢+ ; X); observe that Wloc (⺢+ ; X)→ C(⺢+ ; X). p Thus (10.1) is L -stable (S) for each p ∈ [1, ∞]. Even more is true. If 10.1 is L1 -stable(S), choosing f (t) ≡ x we obtain S(·)x ∈ 1 L (⺢+ ; X) for each x ∈ X, i.e. S(t) is strongly integrable. Conversely, if S(t) is strongly integrable and f˙ ∈ L1 (⺢+ ; X) is a simple function, say f˙ = χAi xi , then u = Sf (0) + χAi ∗ (Sxi )

is well-defined, and |u|1 ≤ |Sf (0)|1 +

|Sxi |1 mes(Ai ) ≤ M (|f (0)| + |f˙|1 ),

where M = sup{|Sx|1 : |x| ≤ 1}. Since simple functions are dense in L1 (⺢+ ; X), this estimate shows that (10.1) is L1 -stable(S). Thus L1 -stability is equivalent to strong integrability of S(t). Similarly, suppose (10.1) admits an integrable integral resolvent R(t); cp. Section 1.6. Then the mild solution of (10.1) for f = a ∗ g, g ∈ Lp (⺢+ ; XA ) is given by t

u(t) = 0

R(t − τ )g(τ )dτ,

t ∈ ⺢+ ,

(10.3)

and so u ∈ Lp (⺢+ ; X). Thus (10.1) is Lp -stable (R) for each p ∈ [1, ∞]. Next let us derive necessary conditions for Lp -stability in terms of Laplace transforms. For this purpose we let as before H(λ) = and

1 (I − a ˆ(λ)A)−1 , λ

K(λ) = a ˆ(λ)(I − a ˆ(λ)A)−1 ,

(10.4)

(10.5)

wherever these operators are defined. Now suppose (10.1) is Lp -stable (R) for some p ∈ [1, ∞]. By p ∈ [1, ∞] we denote the conjugate exponent defined by 1/p + 1/p = 1. Let f = a ∗ g, g = e−μt x, where x ∈ X and μ ∈ ⺓+ are fixed. Then there is a unique solution uμ (·; x) ∈ Lp (⺢+ ; X), and by the closed graph theorem there is a constant C > 0 such that |uμ (·; x)|p ≤ C|g|p = C(pRe μ)−1/p |x|. (10.6)

258

Chapter III. Equations on the Line

Define an operator family {Uμ (t)}t≥0 ⊂ B(X) by means of Uμ (t)x =

t 0

uμ (s; x)ds,

t ≥ 0, x ∈ X;

then Uμ (t) is strongly continuous and |Uμ (t)| ≤ c(μ)t1/p , t ≥ 0. By uniqueness of the solutions of (10.1) we have Auμ (t; x) = uμ (t; Ax) for a.e. t ∈ ⺢+ , and all x ∈ D(A), hence Uμ (t) commutes with A for each t ≥ 0. Taking Laplace transforms, (10.1) implies ˆμ (λ)x = a (I − a ˆ(λ)A)U ˆ(λ)[λ(λ + μ)]−1 x,

x ∈ X,

ˆμ (λ) commutes with A this identity shows that for Re λ sufficiently large. Since U I −a ˆ(λ)A is bijective for all sufficiently large Re λ for which a ˆ(λ) = 0 and ˆμ (λ). a ˆ(λ)(I − a ˆ(λ)A)−1 = λ(λ + μ)U This identity shows as in Section 2.1 that a ˆ(λ) admits a meromorphic extension to ⺓+ and a ˆ(λ) can only vanish if A is bounded; 1/ˆ a(λ) ∈ ρ(A) on ⺓+ . This shows that H(λ) and K(λ) defined by (10.4), (10.5), respectively, exist on ⺓+ and are holomorphic in B(X, XA ). Moreover, with (10.6) we obtain

|ˆ uμ (λ; x)| ≤ |uμ |p (p Re λ)−1/p ≤ C (Re μ)−1/p (Re λ)−1/p |x|,

Re λ, μ > 0,

and this implies with u ˆμ (λ; x)(λ + μ) = K(λ)x

|K(λ)| ≤ C (Re μ)−1/p (Re λ)−1/p |λ + μ|,

Re λ, Re μ > 0.

¯ this inequality results in |K(λ)| ≤ 2C , Re λ > 0, i.e. K(λ) is Choosing μ = λ uniformly bounded in B(X) on ⺓+ . In case p = 1 we even see that K(λ) admits a strongly continuous and uniformly bounded extension to ⺓+ , since u ˆμ (λ; x) is continuous on ⺓+ . If (10.1) is even strongly Lp -stable we obtain by similar reasoning the stronger estimate |AK(λ)| + |K(λ)| ≤ C, Re λ > 0, which implies in particular |λH(λ)| ≤ C on ⺓+ , i.e. Equation (10.1) is parabolic in the sense of Definition 3.1. Let us study the behaviour of K(λ) on the imaginary axis in more detail. Suppose K(λ) is strongly continuous and bounded on ⺓+ , and consider a point iρ, ρ ∈ ⺢, let (λn ) ⊂ ⺓+ , λn → iρ, be such that 1/ˆ a(λn ) = zn → z ∈ ⺓. Then z ∈ ρ(A) and K(λn ) → (z − A)−1 in B(X); if zn → ∞ then, by the boundedness of K(λn ), K(λn ) → 0 strongly. Therefore we see that limλ→iρ a ˆ(λ) = a ˆ(iρ) ∈ ⺓ ∪ {∞} exists for each ρ ∈ ⺢, i.e. a ˆ(λ) admits continuous extension to ⺓+ and the representation (10.1) holds also on ⺓+ . Observe that in case a ˆ(λ) = ∞ for some λ ∈ ⺓+ then A is necessarily invertible. We summarize what we have proved so far in


259

Proposition 10.1 (i) If (10.1) is strongly Lp -stable then (10.1) is parabolic in the sense of Definition 3.1. (ii) If (10.1) admits an integrable integral resolvent R(t) then (10.1) is Lp -stable (R) for each p ∈ [1, ∞]. (iii) If (10.1) is Lp -stable (R) for some p ∈ [1, ∞], then a ˆ(λ) admits meromorphic extension to ⺓+ , 1/ˆ a(λ) ∈ ρ(A), and K(λ) defined by (10.5) is uniformly bounded on ⺓+ ; a ˆ(λ) does not vanish on ⺓+ in case A is unbounded, and can only have poles if A is invertible. (iv) If (10.1) is L1 -stable (R) then a ˆ(λ) admits continuous extension in ⺓ ∪ {∞} to ⺓+ , 1/ˆ a(λ) ∈ ρ(A), and K(λ) defined by (10.5) is strongly continuous on ⺓+ ; a ˆ(λ) = ∞ for some λ ∈ ⺓+ implies 0 ∈ ρ(A). On the other hand, if (10.1) is Lp -stable (S) then a reasoning similar to the one above yields |H(λ)| ≤ M on ⺓+ , and a strongly continuous extension of H(λ) to ⺓+ in case p = 1. Next we consider the behaviour of H(λ) on i ⺢ in case p = 1. If 0 ∈ σ(A) then |z(z − A)−1 | ≥ 1 for all z ∈ ρ(A), hence M≥

1 |(I − a ˆ(λ)A)−1 | ≥ 1/|λ|, |λ|

a contradiction. Therefore, boundedness of H(λ) near zero implies already 0 ∈ ρ(A). Let (λn ) ⊂ ⺓+ , λn → iρ = 0, be such that a ˆ(λn ) → z ∈ ⺓ ∪ {∞}; then H(λn ) → H(iρ) = (I −zA)−1 /iρ in B(X) in case z = 0, ∞, H(λn ) → H(iρ) = 1/iρ strongly in case z = 0, and H(λn ) → H(iρ) = 0 in B(X) in case z = ∞. This shows that a ˆ(λ) admits continuous extension in ⺓ ∪ {∞} to ⺓+ \ {0}. Similarly, for λn → 0 we obtain in case z ∈ ⺓ 0 = lim λn H(λn )x = lim (I − a ˆ(λn )A)−1 x = (I − zA)−1 x, n→∞

n→∞

x ∈ X,

which is impossible, hence a ˆ(λ) → a ˆ(0) = ∞ as λ → 0. But then −A−1 x lim (λˆ a(λ))−1 = lim (λˆ a(λ))−1 (1/ˆ a(λ) − A)−1 x = lim H(λ)x = H(0)x, λ→0

λ→0

λ→0

i.e. λˆ a(λ) → a(∞) as λ → 0 exists and is nonzero but possibly infinity. Let us summarize these results on Lp -stability (S) in Proposition 10.2 (i) If (10.1) admits an integrable and bounded resolvent S(t) then (10.1) is Lp -stable (S) for each p ∈ [1, ∞]. (ii) If (10.1) is Lp -stable (S) for some p ∈ [1, ∞] then a ˆ(λ) admits meromorphic extension to ⺓+ , 1/ˆ a(λ) ∈ ρ(A), and H(λ) defined by (10.4) is uniformly bounded on ⺓+ ; A is invertible, and a ˆ(λ) does not vanish on ⺓+ in case A is unbounded. (iii) If (10.1) is L1 -stable (S) then a ˆ(λ) admits continuous extension to ⺓+ , 1/ˆ a(λ) ∈ ρ(A), and H(λ) is strongly continuous on ⺓+ ; a ˆ(0) = ∞ and the limit limλ→0 λˆ a(λ) = a(∞) exists in ⺓ ∪ {∞} and is nonzero. (iv) S(t) is strongly integrable iff (10.1) is L1 -stable(S).

260


Note that we have the relations ∞ R(t)dt = K(0) = (1/ˆ a(0) − A)−1 ,

(10.7)

0

and

∞ 0

S(t)dt = H(0) = −A−1 /a(∞),

(10.8)

whenever R resp. S are strongly integrable. In general, it is not true that the necessary conditions on H(λ) resp. K(λ) obtained in Proposition 10.2 and 10.1 are also sufficient for the integrability of S(t) resp. R(t), even if their existence is a priori known. There are several counterexamples in the case a ˆ(λ) = 1/λ, i.e. the case of first order differential equations u˙ = Au + g; then H(λ) = K(λ) = (λ − A)−1 ,

i.e. R(t) = S(t) = eAt .

The next subsections deal with the sufficiency of the transform conditions for integrability under various extra assumptions frequently met in applications. At first only equations of scalar type are considered for which the theory is more complete, but later, extensions to nonscalar equations will also be discussed. As before, we concentrate on S(t) which is the more important quantity. 10.2 Parabolic Equations of Scalar Type Before necessary and sufficient conditions for uniform integrability of resolvents for parabolic equations of scalar type are derived we have to recall the notion of a locally analytic function on the right halfplane. ∞ Definition 10.3 Let ⺓∞ + = ⺓+ ∪ {∞} and ϕ : ⺓+ → ⺓. Then (i) ϕ is called locally analytic at a point λ0 ∈ ⺓+ if there are r > 0, n functions kj ∈ BV (⺢+ ), and a holomorphic function ψ : Ω → ⺓, where Ω ⊂ ⺓n+1 is an 1 (λ0 ), . . . , dk n (λ0 )) ∈ Ω, such that open neighbourhood of (λ0 , dk

1 (λ), . . . , dk n (λ)), ϕ(λ) = ψ(λ, dk

λ ∈ Br (λ0 ) ∩ ⺓+ .

(10.9)

(ii) ϕ is called locally analytic at ∞ if there are R > 0, n functions aj ∈ L1 (⺢+ ), m functions kj ∈ BV (⺢+ ) and a holomorphic function ψ : Ω → ⺓, where Ω ⊂ ⺓n+m+1 is open with 0 ∈ Ω, and 1 ϕ(λ) = ψ( , a ˆ1 (λ), . . . , a ˆn (λ), kˆ1 (λ), . . . , kˆm (λ)), λ

λ ∈ ⺓∞ + \ BR (0).

(10.10)

(iii) ϕ is called locally analytic on D ⊂ ⺓∞ + if it is locally analytic at every point of D. Observe that a function ϕ is locally analytic on ⺓+ if and only if is holomorphic on ⺓+ . The number n and the functions kj in Definition 10.3 (i) may vary from


261

point to point, hence there is a lot of flexibility in proving that a given function is locally analytic on ⺓∞ + . Note that locally analytic functions are continuous. The reason for our interest in locally analytic functions is the following result. Lemma 10.1 Suppose ϕ is locally analytic on ⺓∞ + . Then there is a function b ∈ L1 (⺢+ ) such that ϕ(λ) = ϕ(∞) + ˆb(λ)

for all λ ∈ ⺓∞ +.

(10.11)

Lemma 10.1 is an extension of the general Wiener-Levy theorem for the Laplace transform, which is essentially due to Gelfand, Raikhov and Shilov [128]. The proof relies on several results in complex function theory of several variables and cannot be given here; cp. Section 10.7 for further remarks. It is not difficult to show that in the definition of local analyticity at λ0 ∈ ⺓+ , the measures dkj can be replaced by L1 -functions and the explicit dependence on λ can be dropped, by passing to a different holomorphic function ψ0 . In fact, with a ˆ0 (λ) = (λ + 1)−1 we may write 1 , . . . , dk n ) = ψ( 1 − 1, 1 · a 1 , . . . , 1 · a n ) ψ(λ, dk ˆ0 dk ˆ0 dk a ˆ0 a ˆ0 a ˆ0 = ψ0 (ˆ a0 , a ˆ1 , . . . , a ˆn ), where aj = a0 ∗ dkj ∈ L1 (⺢+ ), and ψ0 is analytic in a neighborhood of (ˆ a0 (λ0 ), a ˆ1 (λ0 ), . . . , a ˆn (λ0 )); observe that a ˆ0 (λ0 ) = 0 for λ0 ∈ ⺓+ . Similarly, at ∞ the explicit dependence of ψ on 1/λ, kˆ1 , . . . , kˆm can be dropped since 1/λ = a ˆ0 /(1−ˆ a0 ), j (λ)/λ. and kˆj (λ) = dk Another useful remark is the following. If ϕ is locally analytic on an open set E ⊂ i⺢ and χ ∈ ⺓∞ 0 (⺢) with supp χ ⊂ E then ϕ(iρ)·χ(ρ) is the Fourier transform of a function in L1 (⺢). This can be proved similar to the Paley-Wiener lemma by a localization procedure and the general Wiener-Levy theorem for the Fourier transform. Suppose now that (10.1) admits a strongly integrable resolvent S(t); then for each λ0 ∈ ⺓+ for which a ˆ(λ0 ) = 0 there are x ∈ D(A), x∗ ∈ X ∗ such that ∗ < x, x > = 1 and λ0 < H(λ0 )x, x∗ > = < x, x∗ > = 1. As in Section 2.1 this implies the representation 1 λˆ a(λ)

=

< H(λ)Ax, x∗ > λ < H(λ)x, x∗ > −1

= g(λ, a ˆ1 (λ), a ˆ2 (λ)),

λ ∈ Br (λ0 ) ∩ ⺓+ ,

where a1 (t) = < S(t)Ax, x∗ >, a2 (t) = < S(t)x, x∗ > belong to L1 (⺢+ ) by assumption, and the function g(λ, z1 , z2 ) = z1 /(λz2 − 1) is analytic on a neighborhood of (λ0 , a ˆ1 (λ0 ), a ˆ2 (λ0 )). This shows that the function 1/λˆ a(λ) is locally analytic on ⺓+ \ a ˆ−1 (0). It follows from Proposition 10.2 that a ˆ(λ) = 0 on ⺓+ if A is unbounded and the resolvent S(t) is integrable, since then H(λ) is B(X)-continuous on ⺓+ .

262


We can now prove the following characterization of uniform integrability of analytic resolvents. Theorem 10.1 Suppose S(t) is an analytic resolvent for (10.1), and let A be unbounded. Then S(t) is uniformly integrable if and only if the following conditions hold. (I1) 1/ˆ a(λ) admits a continuous extension in ⺓ to ⺓+ \ {0} which is holomorphic on ⺓+ and 1/ˆ a(λ) ∈ ρ(A) for all λ ∈ ⺓+ , λ = 0; (I2) λˆ a(λ) → a(∞) = 0 as λ → 0, and 0 ∈ ρ(A); (I3) 1/λˆ a(λ) is locally analytic on i⺢. Proof: The necessity of the conditions is clear from Proposition 10.2 and the remarks in front of this theorem. To prove sufficiency recall from Theorem 2.1 that H(λ) as well as 1/ˆ a(λ) admit analytic extension to a sector Σ(ω0 , θ0 + π/2) and that |H(λ)| ≤ M (ω, θ)/|λ − ω| holds on Σ(ω, θ + π/2), ω > ω0 , θ < θ0 . From Cauchy’s theorem we also obtain |H (λ)||λ − ω|2 + |H

(λ)||λ − ω|3 ≤ M (ω, θ),

λ ∈ Σ(ω, θ + π/2).

(I1) and (I2) then show that H(λ) is B(X)-continuous on ⺓+ ∪ Σ((ω, θ + π/2) and bounded, and analytic in the interior of this set. Therefore, by Cauchy’s theorem and (2.10), we obtain the representation 1 S(t) = H(λ)eλt dλ, t ∈ ⺢+ , 2πi Γ where Γ consists of the two halfrays ±i(N + 3) + e±i( 2 +θ) ⺢+ and the interval [−i(N + 3), i(N + 3)], with N > ω cot θ. Choose a C0∞ (⺢)-function ϕ(ρ) ˜ such that ϕ(ρ) ˜ = 1 for |ρ| ≤ N + 1, ϕ(ρ) ˜ =0 for |ρ| ≥ N + 2, 0 ≤ ϕ˜ ≤ 1 elsewhere. Then S(t) decomposes according to π

∞ 1 1 iρt H(iρ)ϕ(ρ)e ˜ dρ + H(λ)ψ(λ)eλt dλ 2π −∞ 2πi Γ = S1 (t) + S2 (t), t ∈ ⺢+ ,

S(t) =

where ψ(iρ) = 1 − ϕ(ρ), ˜ ρ ∈ ⺢, ψ(λ) = 1 for λ ∈ i⺢. Integrating by parts twice in the integral defining S2 (t) we obtain an estimate of the form |S2 (t)| ≤ C(1 + t2 )−1 , t ≥ 0, hence S2 ∈ L1 (⺢+ ; B(X)). Although ϕ˜ ∈ C0∞ (⺢), this trick cannot be applied directly to S1 (t), since we only know that H(iρ) is B(X)-continuous on ⺢, in particular at ρ = 0. To circumvent this difficult we use the Paley-Wiener theorem and local analyticity of 1/λˆ a(λ) on i⺢. For this purpose approximate the function b(ρ) = (iρˆ a(iρ))−1 by some bε ∈ C0∞ (⺢) uniformly on [−(N + 5), N + 5], and consider the operator-valued function Hε (ρ) defined by −1 Hε (ρ) = ϕ(ρ)b ˜ , ε (ρ)(bε (ρ)iρ − A)

ρ ∈ ⺢,


263

where ε ≥ 0 is small enough. Since Hε (ρ) has compact support and is of class C ∞ there follows Hε (ρ) = S˜1ε (ρ) for some S1ε ∈ L1 (⺢; B(X)). We have the identity ϕ(ρ)H(iρ) ˜

= Hε (ρ) + ϕ(ρ)(b(ρ) ˜ − bε (ρ))(bε (ρ)iρ − A)−1 − iρ(b(ρ) − bε (ρ)(bε (ρ)iρ − A)−1 ϕ(ρ)H(iρ), ˜ ρ ∈ ⺢,

hence with ϕ˜1 (ρ) = ϕ(ρ ˜

N +2 ) N +4

ϕ(ρ)H(iρ) ˜ = Hε (ρ) + ϕ(ρ)K ˜ ˜ ε (ρ) − iρKε (ρ)ϕ(ρ)H(iρ),

ρ ∈ ⺢,

(10.12)

where Kε (ρ) = ϕ˜1 (ρ)(b(ρ) − bε (ρ))(bε (ρ)iρ − A)−1 ,

ρ ∈ ⺢.

Since 1/λˆ a(λ) is locally analytic on i⺢, there follows existence of a function ˜ ε (ρ) =Kε (ρ), ρ ∈ ⺢. Thus consider the conRε ∈W 1,1 (⺢; B(X)) such that R volution equation corresponding to (10.12), i.e. S1 = (S1ε + ϕ ∗ Rε ) + R˙ ε ∗ S1

(10.13)

and apply the Paley-Wiener lemma to obtain S1 ∈ L1 (⺢; B(X)); the Paley-Wiener condition is satisfied since

I for ρ ∈ M := supp ϕ˜1 ˜ ˙ I − Rε (ρ) = {(ϕ˜1 (ρ)b(ρ) + (1 − ϕ˜1 (ρ))bε (ρ))iρ − A}(bε (ρ)iρ − A)−1 , ρ ∈ M, is invertible for all ρ ∈ ⺢, provided ε > 0 is chosen small enough. Therefore ϕ(ρ)H(iρ) ˜ is the Fourier transform of some function S1 ∈ L1 (⺢; B(X)), and so the resolvent for (10.1) is uniformly integrable. 2

For an application of Theorem 10.1 let us take up Examples 2.2 again. So let A generate a bounded analytic semigroup in X and let a ∈ L1loc (⺢+ ) be completely monotonic. Then by Corollary 2.4 there is an analytic resolvent for (10.1) which in fact is also bounded. (I1) is trivially satisfied, and (I2) holds if a(∞) = limλ→0 λˆ a(λ) = limt→∞ a(t) > 0 and A is invertible. Since 1/ˆ a(λ) is a Bernstein function we obtain k∞ ˆ 1 = k0 + + k1 (λ), λˆ a(λ) λ

Re λ > 0,

for some k0 , k∞ ≥ 0, k1 ∈ L1loc (⺢+ ) completely monotonic, thanks to Proposition 4.6. a(∞) > 0 then implies k∞ = 0, k1 ∈ L1 (⺢+ ) and so 1/λˆ a(λ) is locally analytic on ⺓∞ , i.e. (I3) holds. We have proved + Corollary 10.1 Suppose A generates an analytic C0 -semigroup, bounded on some sector Σ(0, θ), and is invertible, let a ∈ L1loc (⺢+ ) be completely monotonic and such that a(∞) = limt→∞ a(t) > 0. Then (10.1) admits an uniformly integrable analytic resolvent.

264


Another simple computation shows that the resolvent S(t) from Example 2.1 is uniformly integrable iff β ≥ 1. By the same method of proof, only a slight modification is needed, one obtains a characterization of uniform integrability of resolvents for parabolic equations with 2-regular kernels. Theorem 10.2 Suppose (10.1) is parabolic in the sense of Definition 3.1 and let a(t) be 2-regular. Then S(t) is uniformly integrable if and only if 1/λˆ a(λ) is locally analytic at λ = 0 and 0 ∈ ρ(A). Proof: We only have to prove sufficiency. From Definition 3.1 and 2-regularity there follows by Lemma 8.1 that H(λ) admits a B(X)-continuous extension to ⺓∞ a(λ) = a(∞) = + \ {0}, H(∞) = 0. Since 0 ∈ ρ(A) by assumption and limλ→0 λˆ 0 by local analyticity of 1/λˆ a(λ), we see that H(λ) is also continuous at λ = 0. Therefore we have |H(λ)| ≤ M/(1 + |λ|), Re λ ≥ 0, as well as |λ2 H (λ)| + |λ3 H

(λ)| ≤ M , Re λ > 0, as in the proof of Theorem 3.1, since a(t) is by assumption 2-regular. For fixed x ∈ X, x∗ ∈ X ∗ , the L2 -theory of the Laplace transform therefore yields the representation ∞ 1 ∗ < S(t)x, x > = < H(iρ)x, x∗ > eiρt dρ, a.a. t ∈ ⺢+ . (10.14) 2π −∞ With ϕ˜ as in the proof of Theorem 10.1, N > 0 being arbitrarily fixed, after two integrations by parts (10.14) becomes ∞ 1 ∗ < S(t)x, x > = < ϕ(ρ)H(iρ)x, ˜ x∗ > eiρt dρ 2π −∞ ∞ 1

− < [(1 − ϕ(iρ))H(iρ)] ˜ x, x∗ > eiρt dρ 2πt2 −∞ for a.a. t ∈ ⺢+ . Since both integrals in this formula exist absolutely we may drop x, x∗ to the result ∞ ∞ 1 1 iρt

iρt S(t) = ϕ(ρ)H(iρ)e ˜ dρ − [(1 − ϕ(iρ))H(iρ)] ˜ e dρ 2π −∞ 2πt2 −∞ = S1 (t) + S2 (t), t ∈ ⺢+ . Since S(t) is already known to be bounded by Theorem 3.1 and S1 (t) obviously is, S2 (t) is bounded, but also |S2 (t)| ≤ C/t2 holds, i.e. S2 ∈ L1 (⺢; B(X)). On the other hand, S1 (t) can be treated as in the proof of Theorem 10.1, once it is known that 1/λˆ a(λ) is locally analytic on i⺢. By assumption this is true for λ = 0, and for λ ∈ i⺢, λ = 0, this follows from Lemma 10.2 given below. The proof is complete. 2

The following lemma contains conditions sufficient for local analyticity of the function (λˆ a(λ)))−1 .


265

Lemma 10.2 Suppose a(t) is 1-regular and let ϕ(λ) = 1/(λk a ˆ(λ)) for some k ∈ ⺞0 . Then (i) ϕ(λ) is locally analytic on ⺓+ \ {0}; (ii) if lim|λ|→0 ϕ(λ) = ϕ(0) ∈ ⺓ exists and ϕ (i·) ∈ L1 (−1, 1), then ϕ(λ) is locally analytic at λ = 0; (ii) if lim|λ|→∞ ϕ(λ) = ϕ(∞) ∈ ⺓ exists and ϕ (i·) ∈ L1 (⺢ \ [−1, 1]) then ϕ(λ) is locally analytic at λ = ∞. Proof: (i) From Lemma 8.1 (iv) and (v) there follows the existence of a constant c > 0 and N ∈ ⺞ such that |ˆ a(λ)| ≥ c|λ|−N as |λ| → ∞ and |ˆ a(λ)| ≥ c|λ|N as |λ| → 0. The function ψ(λ) =

λN +1 , (1 + λ)2N +2 a ˆ(λ)

Re λ > 0,

is therefore bounded on ⺓+ , and by 1-regularity of a(t), ψ belongs to H 1 (⺓+ ). Hardy’s inequality (cp. Duren [106]) then yields b ∈ L1 (⺢+ ) such that ˆb = ψ; but this leads to ϕ(λ) =

2N +2 1 ˆb(λ) · (1 + λ) = , λk a ˆ(λ) λN +1+k

λ ∈ ⺓+ \ {0},

i.e. ϕ(λ) is locally analytic on ⺓+ \ {0}. (ii) For the second assertion consider correspondingly ψ(λ) =

1 , (λ + 1)N +1 λk a ˆ(λ)

Re λ > 0;

then again ψ(λ) is bounded on ⺓+ , and ψ belongs to H 1 (⺓+ ), for |ψ (λ)| ≤ c/|λ|2 as |λ| → ∞, and 1 1

|ψ (iρ)|dρ ≤ C + C |ϕ (iρ)|dρ < ∞. −1

−1

Thus ψ(λ) = ˆb(λ) for some b ∈ L1 (⺢+ ) by Hardy’s inequality again, and so ϕ(λ) =

1 = ˆb(λ)(1 + λ)N +1 , λk a ˆ(λ)

λ ∈ ⺓+ ,

i.e. ϕ(λ) is locally analytic on ⺓+ . (iii) is proved similarly by taking ψ(λ) = (

λ N +k+1 1 ) · k , 1+λ λ a ˆ(λ)

Re λ > 0. 2

In the situation of Corollary 3.2, 1/λˆ a(λ) is locally analytic at zero iff a(∞) > 0; in fact, we may write 1 1 2 (λ)), = = f (λ, a ˆ1 (λ), da 2 (λ) λˆ a(λ) λˆ a1 (λ) + da

Re λ ≥ 0,

266


where a1 ∈ L1 (⺢+ ), a2 ∈ BV (⺢+ ), and f (λ, z, w) is analytic near λ0 = 0, z0 = 2 (0) = a(∞) > 0. Therefore, a(∞) > 0 and 0 ∈ ρ(A) imply a ˆ1 (0), w0 = da with Theorem 10.2 that the resolvent S(t) obtained in Corollary 3.2 is uniformly integrable. t Similarly, in the situation of Corollary 3.3 where a(t) = a0 + 0 a1 (s)ds, a0 ≥ 0, ∞ a1 2-monotone, we have a(∞) = a0 + 0 a1 (t)dt > 0, unless a(t) ≡ 0. Setting t t ϕ(t) = a1 (s)ds, ψ(t) = sa1 (s)ds, t > 0, 0

0

with Propositions 3.8 and 3.9 we get 1 1 1 d |ˆ a 1 (iρ)| | (iρˆ a(iρ))−1 |dρ = dρ ≤ C ψ(1/ρ)ϕ(1/ρ)−2 dρ 2 dρ |a + a ˆ (iρ)| 0 1 −1 −1 0 ∞ ∞ −2 2 −2 =C ψ(t)ϕ(t) dt/t ≤ C(ψ(1)ϕ(1) + ψ (t)ϕ(t)−2 dt/t) 1 1 ∞ −2

−2 = C(ψ(1)ϕ(1) + ϕ (t)ϕ(t) dt) ≤ C(ψ(1)ϕ(1)−2 + ϕ(1)−1 ) < ∞. 1

Therefore, 1/λˆ a(λ) is locally analytic at λ = 0, by Lemma 10.2. Theorem 10.2 then shows that the resolvent S(t) obtained in Corollary 3.3 is uniformly integrable if and only if 0 ∈ ρ(A). 10.3 Subordinated Resolvents ∞ Suppose b ∈ L1loc (⺢+ ) satisfies 0 |b(t)|e−βt dt < ∞ for some β ≥ 0, let A be closed linear unbounded and densely defined, and assume there is a resolvent Sb (t) for t v(t) = g(t) + b(t − s)Av(s)ds, t ∈ ⺢+ , (10.15) 0

such that

ωb = limt→∞ t−1 log |Sb (t)| < ∞.

Let c(t) be a completely positive function with associated creep function t k(t) = κ + ωt + k1 (s)ds, t > 0, (10.16) 0

where κ, ω ≥ 0, k1 (t) ≥ 0 nonincreasing with limt→∞ k1 (t) = 0, and let α = ω + k1 (0+). According to Theorem 4.1 there is a resolvent Sa (t) for (10.1), where a ∈ L1loc (⺢+ ) defined by a ˆ(λ) = ˆb(1/ˆ c(λ)),

Re λ sufficiently large ,

(10.17)

is also of exponential growth, by Proposition 4.7. The growth bound of Sa (t) is not greater than ωa , where ωa = cˆ−1 (1/ωb )

if ωb > 1/ω,

ωa = 0 otherwise.

(10.18)


267

In this subsection we want to discuss uniform and strong integrability of Sa (t). Our approach is based on the Paley-Wiener lemma and depends heavily on the structure theorem, Theorem 4.4, for subordinated resolvents; therefore we must 1,1 (0, ∞). The decomposition (4.57) also assume that the function k1 belongs to Wloc shows that Sa (t) can only be expected to be B(X)-measurable if α + 1/κ = ∞, unless we restrict the class of kernels b and operators A, so that Sb (t) is B(X)measurable. This indicates that the two cases α + 1/κ = ∞ and α + 1/κ < ∞ should be treated separately. We do not impose any extra condition on b(t), except for a mild regularity assumption for ˆb(z) at z = ∞. We begin with the case 1/κ + α = ∞. Theorem 10.3 Suppose the function g(z) = z(ˆb(η + z) − ˆb(z))/ˆb(z) is locally analytic at ∞, for some η > max{ωb , β}, and let 1/κ + α = ∞. Then Sa (t) is uniformly integrable if and only if Conditions (I1), (I2), (I3) of Theorem 10.1 are satisfied. Proof: The necessity of (I1), (I2), (I3) is already clear. To prove the sufficiency, fix any η > max{ωb , β} such that g(z) is locally analytic at ∞. To apply the Paley-Wiener lemma Theorem 0.7 we have to have an object Sη ∈ L1 (⺢+ ; B(X)) to start with. Define Hη (λ) by means of c(λ)) Hη (λ) = z −1 (I − ˆb(z)A)−1 ◦ (η + 1/ˆ ∞ = Sb (τ )e−ητ e−τ /ˆc(λ) dτ, Re λ ≥ 0;

(10.19)

0

then Hη (λ) = Sˆη (λ), Re λ ≥ 0, where ∞ Sb (τ )e−ητ wt (t, τ )dτ, Sη (t) =

t ≥ 0,

0

(10.20)

belongs to L1 (⺢+ ; B(X)) since α+1/κ = ∞ by assumption; thanks to Propositions 4.9 and 4.10. Next one verifies easily the identity H(λ) = ϕη (λ)Hη (λ) − ψη (λ)Hη (λ)H(λ),

Re λ > 0,

(10.21)

where c(λ))ˆb(η + 1/ˆ c(λ))/λˆ a(λ), ϕη (λ) = (η + 1/ˆ

Re λ > 0,

and c(λ))(ˆb(η + 1/ˆ c(λ)) − ˆb(1/ˆ c(λ))/ˆb(1/ˆ c(λ)), ψη (λ) = (η + 1/ˆ

Re λ > 0.

Then the Paley-Wiener condition is implied by (I1), (I2) since I + ψη Hη

= [(I − ˆb(η + 1/ˆ c)A) + ˆb(η + 1/ˆ c)/ˆb(1/ˆ c) − I] · (I − ˆb(η + 1/ˆ c)A)−1 = ˆb(η + 1/ˆ c)[1/ˆ a − A](I − ˆb(η + 1/ˆ c)A)−1

268


is invertible for each Re λ ≥ 0 and (I + ψη Hη )−1 = (1/ˆb(η + 1/ˆ c) − A)(1/ˆ a − A)−1 ; observe that ˆb(η + 1/ˆ c) is well-defined and nonzero by the choice of η. Thus to apply Theorem 0.7 it remains to show that ϕη and ψη are Laplace transforms of integrable functions on the halfline; this will be achieved by showing that these functions are locally analytic on ⺓∞ + and then applying Lemma 10.1. ˆ To prove this observe that b(η + 1/ˆ c) = a ˆη for some function aη ∈ L1 (⺢+ ), by Proposition 4.7 and the choice of η. The function 1/ˆ c(λ) is a Bernstein function which is locally analytic on ⺓+ by decomposition (4.9). Together with (I3) these facts imply that ϕη (λ) and ψη (λ) are locally analytic on ⺓+ . The difficult part is to show that ψη (λ) is locally analytic at ∞, and here we need the hypothesis about g(z). By assumption g(z) is locally analytic at infinity, hence gη (z) = g(z) + ηg(z)/z is as well, and so by the remarks following Lemma 10.1 gη (z) = ψ(ˆ a1 (z), . . . , a ˆn (z)), Re z ≥ 0, |z| > R, for some ψ analytic near 0, aj ∈ L1 (⺢+ ). But then by Proposition 4.7 ψη (λ) = =

gη (1/ˆ c(λ)) = ψ(ˆ a1 (1/ˆ c(λ)), . . . , a ˆn (1/ˆ c(λ))) ˆ ˆ ψ(b1 (λ), . . . , bn (λ)), Re λ ≥ 0, |λ| ≥ r

with bj ∈ L1 (⺢+ ) and r sufficiently large; hence ψη (λ) is locally analytic at ∞. Finally, to show that ϕη (λ) is locally analytic at ∞, write ϕη (λ) =

1 η 1 ψη (λ) + + , λ λ λˆ c(λ)

Re λ > 0,

and observe that 1/λˆ c(λ) is locally analytic at ∞ by (4.9) again. 2

As an example where g(z) = z(ˆb(η +z)− ˆb(z))/ˆb(z) is locally analytic at ∞, choose ˆb(z) = z −γ , i.e. b(t) = tγ−1 /Γ(γ), where γ > 0. Then g(z) = z((

z γ η γ ) − 1) = z((1 − ) − 1), η+z η+z

which is even analytic at ∞; in particular the cases of Theorem 4.2 and 4.3, i.e. b(t) = 1, resp. b(t) = t are covered. Next we consider the case 1/κ + α < ∞. Theorem 10.4 Suppose the function g(z, w) = w−1 (ˆb(z)/ˆb(z+zw)−1) is analytic at (∞, 0), let 1/κ + α < ∞ and assume α > ωb . Then Sa (t) is integrable if and only if (I1), (I2), (I3) of Theorem 10.1 are satisfied. Sa (t) is uniformly integrable if and only if Sa (t) is integrable and Sb (t) is B(X)-measurable on ⺢+ . Proof: The second assertion is clear from the structure theorem, Theorem 4.4; we therefore concentrate on the sufficiency part of the first assertion. To apply again


269

a Paley-Wiener argument, we start with the ‘bad’ part of S(t), which by (4.57) is S1 (t) = Sa (t) − S0 (t). S1 (t) is integrable since α > ωb by assumption, and its Laplace transform is given by H1 (λ) = Sˆ1 (λ) = κ(κλ + α)−1 (I − ˆb(κλ + α)A)−1 ,

Re λ ≥ 0;

(10.22)

w.o.l.g. we may assume α > 0 in the sequel, since otherwise S0 (t) ≡ 0, and the assertion is then trivial. A simple computation yields the following identity for H0 (λ) = Sˆ0 (λ). H0 (λ) = ϕ1 (λ)H1 (λ) + ϕ0 (λ)H1 (λ)2 + ϕ0 (λ)H1 (λ)H0 (λ),

Re λ ≥ 0, (10.23)

where ϕ1 (λ) = (1 + α/κλ)ˆb(α + κλ)/ˆ a(λ) − 1,

Re λ ≥ 0,

ϕ0 (λ) = (λ + α/κ)(1 − ˆb(α + κλ)/ˆ a(λ)),

Re λ ≥ 0.

and As in the proof of Theorem 10.3 it is easy to check by (I1), (I2) that the PaleyWiener condition is satisfied, and so by Theorem 0.7 we obtain S0 ∈ L1 (⺢+ ; B(X)) once we have shown that ϕ0 (λ)H1 (λ) and ϕ1 (λ)H1 (λ) are Laplace transforms of functions from L1 (⺢+ ; B(X)). This will be achieved by a local analyticity argument, as in the proof of Theorem 10.3. Since ˆb(z) is meromorphic for Re z > ωb with poles which only may cluster at the vertical line Re z = ωb , ˆb(α + κλ) is holomorphic on ⺓+ , except for finitely many poles λ1 , . . . , λn , say; therefore, by (I3), ϕ1 (λ) and ϕ0 (λ) are locally analytic on ⺓+ \ {λ1 , . . . , λn }. Since g(z, w) is analytic at (∞, 0) we also see that ˆ 1ˆ k˙ 1 (λ) ϕ0 (λ) = − k˙ 1 (λ)g(α + κλ, ), κ α + κλ

Re λ > 0,

is locally analytic at ∞, and so 1 α ϕ1 (λ) = − ϕ0 (λ) − , λ κλ

Re λ > 0,

is as well, and ϕ1 (∞) = ϕ0 (∞) = 0. Thus if ˆb(z) has no poles in Re λ ≥ α, ϕj (λ) = ˆbj (λ), Re λ ≥ 0, for some functions bj ∈ L1 (⺢+ ) by Lemma 10.1, and so ϕj (λ)H1 (λ) is the Laplace transform of an element of L1 (⺢+ ; B(X)). In case ˆb(z) has poles, then H1 (λ) has zeros at these points of the same order, and so the poles ‘drop out’, and one obtains the same conclusion as before. The details of this pole cancelling are left to the reader. 2

Observe that for ˆb(z) = z −γ , γ > 0, one obtains g(z, w) =

1 ((1 + w)α − 1), w

270


and so g obviously is analytic at (∞, 0); this shows that again the cases of Theorem 4.2 and 4.3 are covered. Due to their importance in applications we discuss the cases b(t) = 1 and b(t) = t now separately. So let A generate a C0 -semigroup in X with growth bound ω0 (A), and let b(t) = 1, while c(t) and k(t) are as above. Then Theorems 10.3 and 10.4 apply whenever α > ω0 (A), hence in this case only conditions (I1), (I2), (I3) from Theorem 10.1 have to be checked. For this purpose observe that 1/λˆ c(λ) = κ + kˆ1 + k∞ /λ → 1/c(∞) = ∞,

as λ → 0,

if and only if k∞ = 0 and k1 ∈ L1 (⺢+ ), i.e. k(t) is bounded. If this is the case then 1/λˆ c(λ) obviously is locally analytic on ⺓+ , so (I3) holds, and 1/ˆ c(λ) admits continuous extension in ⺓ to ⺓+ \ {0}.

Ω 0

ω0 (Α)

α

σ (Α)

Figure 10.1: Illustration of Corollary 10.2

Therefore Theorems 10.3 and 10.4 imply Corollary 10.2 Let A generate a C0 -semigroup in X with growth bound ω0 (A), let c(t) be a completely positive function with associated creep function k(t) ∈ 2,1 Wloc (0, ∞) and characteristic numbers κ and α. Define Ω = {1/ˆ c(λ) : λ ∈ ⺓+ , λ = 0}, and assume in addition α > ω0 (A). Then the resolvent S(t) of (10.1) with a(t) = c(t) is integrable if and only if 0 ∈ ρ(A),

k(∞) < ∞,

Ω ∩ σ(A) = ∅.

(10.24)

S(t) is uniformly integrable iff in addition κ = 0 or α = ∞ or eAt is B(X)continuous for t > 0. Consequently, if ω0 (A) < 0 then S(t) will be integrable iff k(∞) < ∞; in the limiting case ω0 (A) = 0, S(t) is integrable iff A is invertible and k(∞) < ∞.


271

Observe that the requirement α > ω0 (A) is trivially satisfied in case ω0 (A) < 0, and for ω0 (A) = 0 it simply means k1 ≡ 0. For the second order subordination b(t) = t, suppose A generates a cosine family Co(t) in X with growth bound ω1 (A), and according to the applications diccussed in Sections 5 and 9 let a(t) be a creep function with log a1 (t) convex; cp. Theorem 4.3 (ii). Then the characteristic numbers of the creep function k(t) associated with the Bernstein function ϕ(λ) = 1/ a ˆ(λ) are κ = 1/ a∞ + a1 (0+) 2,1 if a0 = 0, κ = 0 if a0 > 0, ω = 0, α = −a˙ 1 (0+)κ3 /2; note that k ∈ Wloc (0, ∞), by convexity of a1 (t). To verify (I1)∼(I3) observe that λˆ a(λ) → a(∞) = 0 as λ → 0 is always satisfied; a(∞) is finite in the ‘fluid case’ infinite otherwise. Local analyticity of 1/λˆ a(λ) on ⺓+ is obtained as in the remarks following the proof of Lemma 10.2, referring to the situation of Corollary 3.3. Therefore Theorems 10.3 and 10.4 yield the following result. Corollary 10.3 Let A generate a cosine family Co(t) in X with growth bound ω1 (A), let A be unbounded, a(t) be a creep function with log a1 (t) convex, and let κ and α denote the characteristic numbers of the Bernstein function ϕ(λ) = 1/ a ˆ(λ). Define Ω = {1/ˆ a(λ) : λ ∈ ⺓+ }, and assume in addition α > ω1 (A). Then the resolvent S(t) of (10.1) is integrable if and only if 0 ∈ ρ(A) and Ω ∩ σ(A) = ∅. (10.25) S(t) is uniformly integrable iff in addition κ = 0 or α = ∞. Observe that the cosine family generated by A is never B(X)-measurable, in case A is unbounded, and ω1 (A) ≥ 0. In case ω1 (A) = 0 the situation is particularly nice. In fact, unless a(t) ≡ a∞ t, the restriction α > ω1 (A) is always satisfied, and by Proposition 1.5 we have σ(A) ⊂ (−∞, 0]. On the other hand, 1/ˆ a(λ) ∈ (−∞, 0] iff λ = iρ, for some ρ ∈ ⺢, and λˆ a(λ) ∈ i ⺢; but this is impossible since it would require Re (iρˆ a(iρ)) = a0 + Re a1 (iρ) ≤ 0, −ρIm (iρˆ a(iρ)) = a∞ − ρIm a1 (iρ) = 0. Therefore in case ω1 (A) = 0, the resolvent S(t) is integrable iff 0 ∈ ρ(A) and a(t) ≡ a∞ t. The latter excludes only the ‘purely elastic’ case where S(t) =Co(t/κ), which is never integrable. This result exhibits a rather striking phenomenon of dynamics with memory of this type, e.g. in viscoelasticity problems involving only one kernel of the type considered in Corollary 10.3. The presence of a whatsoever small memory, i.e. a(t) ≡ a∞ t, stabilizes all frequencies except 0. Stability is obtained if in addition the stationary problem is well-posed, i.e. if A is invertible. When considering specific examples, the latter is usually satisfied if the domain under consideration is bounded, but is also valid for several classes of unbounded domains.

272


Let us point out the sharp difference between the ‘fluid case’ a∞ = 0, a1 ∈ L1 (⺢+ ), and that of a solid a∞ > 0. If S(t) is integrable then ∞ S(t)dt = H(0) = −A−1 /a(∞), 0

hence we obtain H(0) = 0 in the ‘solid case’, while H(0) = 0 for ‘fluids’. For further remarks see Section 10.6, and for applications of the foregoing results cp. Section 13. 10.4 Strong Integrability in Hilbert Spaces Let X be a Hilbert space and suppose (10.1) admits an exponentially bounded resolvent S(t) in X. If the kernel a(t) is in addition 1-regular, strong integrability of the resolvent can be characterized in terms of the Laplace transform H(λ) of S(t). The result reads as follows. Theorem 10.5 Suppose (10.1) admits a resolvent S(t) with finite growth bound ω0 (S) < ∞ in the Hilbert space X, and let a(t) be 1-regular. Then S(t) is strongly integrable if and only if (I1), (I2), (I3) of Theorem 10.1 hold and H(λ) is uniformly ¯ +. bounded on ⺓ Proof: Since strong integrability of S(t) implies L1 -stability (S) of (10.1), the necessity part follows from Proposition 10.2 and the remarks in front of Theorem 10.1. To prove sufficiency, choose a cut-off function ϕ(ρ), ˜ i.e. ϕ(ρ) ˜ = 1 for |ρ| ≤ 1, ϕ(ρ) ˜ = 0 for |ρ| ≥ 1, and ϕ˜ ∈ C0∞ (⺢). Then as in the proof of Theorem 10.1 there is S0 ∈ L1 (⺢; B(X)) such that S˜0 (ρ) = ϕ(ρ)H(iρ), ˜

ρ ∈ ⺢.

Therefore it remains to show that for each x ∈ X the function (1 − ϕ(ρ))H(iρ)x ˜ is the Fourier transform of a function in L1 (⺢; X). So let x ∈ X be fixed and put χ(ρ) = 1 − ϕ(ρ). ˜ Choose ω > ω0 (S); then f (t) = e−ωt e0 (t)S(t)x belongs to 2 L (⺢; X), hence by Parseval’s theorem, f˜(ρ) = H(iρ + ω)x is square integrable. The identity ω x H(iρ)x = H(iρ + ω)x + H(iρ)x + h(ρ)H(iρ)(H(ω + iρ)x − ), ω + iρ ω + iρ where h(ρ) = iρ(ˆ a(iρ)/ˆ a(ω + iρ) − 1), then implies χ(·)H(i·)x ∈ L2 (⺢; X) as well since H(iρ) is bounded by assumption, provided h(ρ) is bounded for |ρ| ≥ 1. To prove the latter we employ 1-regularity of a(t). As in the proof of Lemma 8.1 there is a holomorphic function g(λ), Re λ > 0, such that exp (g(λ)) = a ˆ(λ), and |g (λ)| ≤ c/|λ|, Re λ > 0. Then ω d h(ρ) = iρ exp(g(iρ) − g(iρ + s))ds ds 0 ω s = −iρ exp(− g (iρ + τ )dτ )g (iρ + s)ds, 0

0

10. Integrability of Resolvents hence for |ρ| ≥ 1 we obtain |h(ρ)| ≤

ω

273

exp(

0

s 0

|g (iρ + τ )|dτ )|ρ||g (iρ + s)ds

≤ cω exp(cω) < ∞. Thus χ(·)H(i·)x ∈ L2 (⺢; X) and so by Parseval’s theorem there is a function u ∈ L2 (⺢; X) with u ˜(ρ) = χ(ρ)H(iρ)x for a.a. ρ ∈ ⺢. On the other hand, u ˜ (ρ)

= χ (ρ)H(iρ)x + iχ(ρ)H (iρ)x = χ (ρ)H(iρ)x + iχ(ρ)(−H(iρ)x/iρ + H(iρ)iρ

a ˆ (iρ) x (H(iρ)x − )) a ˆ (iρ) iρ

also belongs to L2 (⺢; X) since H(λ) is bounded, χ(ρ) = 0 for |ρ| ≤ 1, and a is 1-regular; see Lemma 8.1. Therefore, by Parseval’s theorem there is v ∈ L2 (⺢; X) such that v˜(ρ) = u ˜ (ρ) for a.a ρ ∈ ⺢. But this implies tu(t) = −v(t), hence |u|1 =

1

−1

|u(t)|dt +

The proof is complete.

|t|≥1

|t||u(t)|

dt √ ≤ 2(|u|2 + |tu|2 ) < ∞. |t|

2

Since D(A∗ ) is dense in X, under the assumptions of Theorem 10.5, S ∗ (t) is the resolvent for (10.1) with A replaced by A∗ , and a(t) replaced by a(t). Theorem 10.5 then shows in case ω0 (S) < ∞ and a(t) 1-regular that with S(t) also S ∗ (t) is strongly integrable. This observation together with the vector-valued Riesz-Thorin interpolation theorem yields Corollary 10.4 Suppose (10.1) admits a resolvent S(t) with finite growth bound ω0 (S) < ∞ in the Hilbert space X, and let a(t) be 1-regular. If S(t) is strongly integrable then (10.1) is Lp -stable (S) for each p ∈ [1, ∞]. Proof: We already know that strong integrability of S(t) implies L1 -stability (S) of (10.1). Since the remark in front of Corollary 10.2 yields also strong integrability of S ∗ (t) we obtain for u(t) = (S ∗ f )(t) the estimate |(u(t), x∗ )| ≤

t 0

|(f (τ ), S ∗ (t − τ )x∗ )|dτ ≤ |f |∞ · |S ∗ (·)x∗ |1

for each f ∈ L∞ (⺢+ ; X), x∗ ∈ X, t ∈ ⺢+ . Thus there is a constant C > 0 such that |u|∞ ≤ |f |∞ , i.e. (10.1) L∞ -stable (S). Interpolation then yields Lp -stability (S). 2

Let us briefy discuss strong integrability of the resolvent S(t) obtained in Corollary 1.2.

274


Corollary 10.5 Let X be a Hilbert space, a ∈ BVloc (⺢+ ) be 1-regular and such that da(t) is of positive type. Then the resolvent S(t) of (10.1) is strongly integrable for every negative definite selfadjoint operator A in X if and only if the following conditions hold. (i) Re da(iρ) > 0 for all ρ ∈ ⺢, ρ = 0; (ii) lim|ρ|→∞ Im da(iρ)/[|ρ| Re da(iρ)] , 0

t ∈ ⺢+ , w ∈ V,

(10.26)

where α : ⺢+ × V × V → ⺓ is such that α(t, ·, ·) is a sesquilinear form on V for ∗ each t > 0, α(0, ·, ·) = 0, and let α(t, ·, ·) be represented by A(t) ∈ B(V, V ) by means of α(t, u, v) = − < u, A(t)v >, t > 0, u, v ∈ V. (10.27) Since Corollary 10.3 is the special case α(t, u, v) = a(t)(A1/2 u, A1/2 v) in this setting, it becomes apparent that for strong integrability of the resolvent S(t) for (10.26) frequency domain conditions reflecting (i) and (ii) of Corollary 10.3 will be needed, as well as some regularity assumptions on the form α(t, u, v) which replace ˙ (iii). We first concentrate on the solid case which corresponds to A(t) → A∞ = 0 as t → ∞. Theorem 10.6 Let α : ⺢+ × V × V → ⺓ satisfy (V1) and (V2) of Theorem 6.5 with α1 (t) bounded, and let α be η-coercive for some η > 0. In addition assume 2,1 (V4) α(·, u, v) ∈ Wloc (0, ∞), α(0, u, v) = 0, for each u, v ∈ V , and |¨ α(t, u, v) − α(s, ¨ u, v)| ≤ (α2 (s) − α2 (t))||u|| ||v||, for all t > s > 0, u, v ∈ V, ∞ where α2 (t) is nonincreasing and such that − 0 tdα2 (t) < ∞; (V5) lim|ρ|→∞ | Re α(iρ, ˆ u, u)/[ρ Im α(iρ, ˆ u, u)]| ≤ φ∞ < ∞ for each u ∈ V ;


275

(S) For each u, v ∈ V , α∞ (u, v) = limt→∞ α(t, ˙ u, v) exists, and with some constant γ∞ > 0 Re α∞ (u, u) ≥ γ∞ ||u||2 , for all u ∈ V. Then the resolvent S(t) for (10.26) and its integral R(t) = (1 ∗ S)(t) satisfy ∗ ˙ ¨ (i) v ∈ V ⇒ S(·)x ∈ L1 (⺢+ ; V ), S(·)v ∈ L1 (⺢+ ; H), S(·)v ∈ L1 (⺢+ ; V ); ∗ ˙ (ii) x ∈ H ⇒ S(·)x ∈ L1 (⺢+ ; H), S(·)x ∈ L1 (⺢+ ; V ), R(·)x ∈ L1 (⺢+ ; V ); ∗ ∗ 1 (iii) u ∈ V ⇒ S(·)u ∈ L (⺢+ ; V ), R(·)u ∈ L1 (⺢+ ; H), and e−ηt ∗ R(·)u ∈ L1 (⺢+ ; V ). Observe that η-coercivity together with (S), (V2), and α1 bounded imply coercivity of α, hence Theorem 6.5 applies, and consequently the exponent 1 in (i), (ii), (iii) of Theorem 10.6 can be replaced by ∞, hence by p ∈ [1, ∞] by interpolation. Since the adjoint form α∗ (t, u, v) = α(t, u, v) also satisfies the assumptions of Theorem 10.6, the argument in the proof of Corollary 10.2 may be applied again, and so (10.26) is Lp -stable (S) for each p ∈ [1, ∞]. Proof of Theorem 10.6: The argument is basically the same as in the proof of Theorem 10.5, therefore we only give an outline here, emphasizing the necessary modifications. ∗ (a) We first show how to obtain the boundedness of H(λ) in B(V ), B(H), ∗ ∗ B(V ), of H(λ)/λ in B(V , H), B(H, V ), and of H(λ)/(λ(λ +η)) in B(V , V ) which are necessary for (i), (ii), (iii) to be valid. (10.26) implies the identity ¯ H(λ)x, H(λ)x) = < H(λ)x, x >, ˆ˙ λ, λ|H(λ)x|2 + α(

(10.28)

∗

valid for Re λ > 0 and x ∈ V . Taking real parts in (10.28), by η-coerciveness we obtain ηγ||H(λ)x|| ≤ ((η + σ)2 + ρ2 )||x||∗ ,

∗

Re λ > 0, x ∈ V ,

(10.29)

˙ (λ) = λ2 t ˙ ¨ ˙ ˙ − λ2 A A(λ) = td A(λ) + 2d A(λ) − A(0),

(10.30)

where λ = σ + iρ. We have

˙ (λ)| ∗ hence |λ2 A B(V,V ) ≤ M since −

∞ 0

tdα2 (t) < ∞ and α1 (t) is bounded. With

˙ (λ))H(λ), H (λ) = −H(λ)(I − A

Re λ > 0,

(10.31) ∗

(10.29) and (10.30) then imply the existence of H(iρ) = limλ→iρ H(λ) in B(V , V ) ˙ ˙ for each ρ = 0; observe that A(λ) = d A(λ)/λ admits continuous extension to ¯ + \ {0}, by boundedness of α1 (t). η-coerciveness and (10.28) for λ = iρ then ⺓ yield ηγ||H(iρ)(η + iρ)−1 x||2 ≤ | < H(iρ)x, x > |,

∗

ρ = 0, x ∈ V ,

(10.32)

276


while decomposing (10.28) into real and imaginary parts, (V5) implies |ρ||H(iρ)x|2 ≤ c(1 + |ρ|)| < H(iρ)x, x > |,

∗

ρ = 0, x ∈ V .

(10.33)

Finally (S) gives ||H(iρ)x|| ≤ c|ρ| ||x||∗ ,

∗

ρ = 0, x ∈ V ;

(10.34)

in particular H(0) = limλ→0 H(λ) = 0. From (10.32), (10.33), (10.34) one easily obtains the boundedness of |H(iρ)|B(H) etc., and then by the maximum principle ¯ +. also on ⺓ (b) As in the proof of Theorem 10.5 consider the identity H(iρ)x

˙ = H(iρ + η)x + ηH(iρ)H(iρ + η)x + H(iρ)(A(iρ) ˙ −A(iρ + η))H(iρ + η)x

(10.35)

∗

where x ∈ V , ρ ∈ ⺢. (V4) and ˙ ˙ ˙ ¨ ˙ ¨ ¨ + η), + η A(0) + 2η A(iρ) + d A(iρ) − d A(iρ (iρ + η)2 (A(iρ) − A(iρ + η)) = η 2 A(iρ) imply ˙ ˙ |(iρ + η)2 (A(iρ) − A(iρ + η))|B(V,V ∗ ) ≤ C(1 + 1/|ρ|),

ρ = 0,

∞ for − 0 tdα2 (t) < ∞ and α1 is bounded. Since e.g. e−ηt S(t)x ∈ L2 (⺢+ ; V ) for x ∈ V , by Parseval’s theorem, we obtain H(iρ + η)x ∈ L2 (⺢+ ; V ), hence H(iρ)x ∈ L2 (⺢+ ; V ) by (10.35) and (a), and so by Parseval’s theorem also S(t)x ∈ L2 (⺢+ ; V ). By similar arguments the assertions of the theorem follow with exponent 2 instead of 1. (c) To obtain e.g. tS(t)x ∈ L2 (⺢+ ; V ) consider (10.31) for λ = iρ. Then for x ∈ V one deduces from (10.31) ||H (iρ)x|| ≤ |H(iρ)|B(V ) ||H(iρ)x|| + |H(iρ)/ρ2 |B(V ∗ ,V ) M ||H(iρ)x||, hence H (iρ)x ∈ L2 (⺢+ ; V ), by the boundedness properties of H(iρ), and by (10.34). Parseval’s theorem gives tS(t)x ∈ L2 (⺢+ ; V ) for each x ∈ V , which together with (b) yields the strong integrability of S(t) in V . The remaining assertions follow by similar arguments. 2

By the same type of ∞arguments one can prove the following result, in which the ˙ ∗ case of a fluid, i.e. 0 |A(t)| B(V,V ) dt < ∞ is considered. Corollary 10.6 Let the assumptions of Theorem 10.6 be satisfied, with the exception that (S) is replaced by (F) |α(t, ˙ u, v)| ≤ α3 (t)||u|| ||v||, t > 0, u, v ∈ V , where α3 ∈ L1 (⺢+ ). ˙ ¨ Then the assertions of Theorem 10.6 for S(t), S(t), S(t) remain valid.


277

Observe that in the fluid case one cannot expect integrability of R(t) since ∞ S(t)dt = H(0) = −A(∞)−1 = 0. 0

Here the difference between the solid and the fluid case becomes apparent, again. 10.5 Nonscalar Parabolic Problems Next we consider integrability properties of the resolvent S(t) for the nonscalar parabolic problem t u(t) = f (t) + A(t − τ )u(τ )dτ, t ≥ 0, (10.36) 0

in the Banach space X, where A ∈ L1loc (⺢+ ; B(Y, X)) is of subexponential growth, d

and Y → X. If (10.36) is (weakly) parabolic and A(t) is (weakly) 1-regular, see Definition 7.3, then by Theorem 7.2, the (weak) resolvent S(t) for (10.36) exists and is bounded in (B(X)) B(X) ∩ B(Y ). If S(t) is integrable in X, by continuity ˆ in B(X) it is also uniformly integrable, and so H(λ) = S(λ) is B(X)-continuous ˆ on ⺓+ . In particular, λH(λ) → 0 as λ → 0, hence λA(λ) is invertible near λ = 0 and −1 −1 ˆ ˆ H(0) = lim (λ − λA(λ)) = − lim (λA(λ)) (10.37) λ→0

λ→0

exists in B(X). Similarly, if S(t) is integrable in Y then (10.37) also holds in B(Y ). For the sufficiency of (10.37) stronger assumptions on the regularity of A(t) and ˆ on the behavior of λA(λ) near λ = 0 are needed. Theorem 10.7 Suppose (10.36) is weakly parabolic, A(t) weakly 2-regular, and let S(t) denote the weak reolvent for (10.36). In addition assume that there is η > 0 −1 ˆ ˆ such that λA(λ) is invertible in X for λ ∈ ⺓+ ∩ Bη (0) and F (λ) = (λA(λ)) ∈ B(X, Y ) satisfies lim F (λ) = F (0) exists in B(X), (10.38) λ→0 η η {|F (iρ)|B(X) +|F (iρ)−F (0)|B(X) /|ρ|}dρ+ |ρ||dF (iρ)|B(X) < ∞. (10.39) −η

−η

1

Then S ∈ L (⺢+ ; B(X)). If (10.36) is parabolic, A(t) 2-regular, and (10.38), (10.39) hold also in B(Y ) then S ∈ L1 (⺢+ ; B(Y )). ˆ Proof: (a) Suppose A(t) is weakly 2-regular. We first show that A(λ) admits a B(Y, X)-continuous extension to ⺓+ \ {0} which is such that (10.39) makes sense. For this purpose observe first that ˆ : λ ∈ ⺓+ , |λ| ≥ r}, ϕ0 (r) = sup{|A(λ)|

r > 0,

is finite, nonincreasing, and limr→∞ ϕ(r) = 0; this follows from 1-regularity by means of Gronwall’s inequality. By 2-regularity we then also have with ϕk (r) = sup{|Aˆ(k) (λ)| : λ ∈ ⺓+ , |λ| ≥ r},

r > 0, k = 1, 2,

278


the estimates ϕk (r) ≤ (1 + ϕ0 (r))/rk ,

r > 0, k = 1, 2.

ˆ Therefore A(λ) and Aˆ (λ) are uniformly Lipschitz on ⺓+ \ Bε (0) for each ε > 0, hence admit extension to ⺓+ \ {0}, and ˆ A(i·) ∈ C01 (⺢ \ {0}; B(Y, X)),

Aˆ (i·) ∈ BVloc (⺢ \ {0}; B(Y, X)),

as well as ˆ |A(iρ)| ≤ ϕ0 (|ρ|), |Aˆ (iρ)| ≤ ϕ1 (|ρ|), |dAˆ (iρ)| ≤ ϕ2 (|ρ|)dρ,

ρ = 0.

Therefore, by (10.38) we see that F (iρ) is of class C 1 for 0 < |ρ| ≤ η and F (i·) ∈ BVloc ([−η, 0) ∪ (0, η]; B(X)); thus (10.39) makes sense. (b) From (a) and weak parabolicity it now follows easily that H(λ) admits continuous extension in B(X) to ⺓+ \ {0}; with ˆ H (λ) = H(λ)(A(λ) + λAˆ (λ) − I)H(λ) and H

(λ) = +

H(λ)(2Aˆ (λ) + λAˆ

(λ))H(λ) ˆ ˆ 2H(λ)(A(λ) + λAˆ (λ) − I)H(λ)(A(λ) + λAˆ (λ) − I)H(λ),

there follows H(i·) ∈ C 1 (⺢ \ {0}; B(X)), H (i·) ∈ BVloc (⺢ \ {0}; B(X)), as well as the estimates |ρ||H(iρ)| + |ρ|2 |H (iρ)| ≤ M,

|ρ|3 |dH (iρ)| ≤ M dρ, ρ = 0.

Furthermore, (10.38) shows the existence of limλ→0 H(λ) = −F (0), for −1 ˆ H(λ) = (λ − λA(λ))) = −F (λ)(I − λF (λ))−1 ,

|λ| < η;

in particular, H(λ) is bounded in B(X) near λ = 0. Finally, (10.39) implies

δ

−δ

{|H (iρ)|B(X) + |H(iρ) − H(0)|B(X) /|ρ|}dρ +

δ −δ

|ρ||dH (iρ)| < ∞,

where 0 < δ ≤ η is chosen small enough, e.g. such that δ|F (λ)| ≤ 1/2 for |λ| ≤ η; observe that by 2-regularity and (10.38), F (λ), λF (λ), λ2 F

(λ) are bounded for λ ∈ ⺓+ ∩ Bη (0). (c) To prove integrability of S(t) in B(X) we begin with the representation tS(t) =

1 2π

∞ −∞

H (iρ)eiρt dρ,

t > 0,


279

which follows from the complex inversion formula of the Laplace-transform; the integral is absolutely covergent by the properties of H (iρ) derived in (b). Integrating by parts for |ρ| ≥ 1/t this formula becomes 1/t 1 1 S(t) = H (iρ)(eiρt − 1)dρ + (H(i/t) − H(−i/t)) 2πt −1/t 2πti 1 1

−i

i + (H (−i/t)e − H (i/t)e ) − eiρt dH (iρ) 2iπt2 2iπt2 |ρ|>1/t t ≥ 1.

= S1 (t) + S2 (t) + S3 (t) + S4 (t),

We estimate the corresponding integrals separately. ∞ 1 s π |S1 (t)|B(X) dt ≤ |H (iρ)|B(X) | sin(ρ/2s)|dρds/s 1

0

−s 1

(

1

=

−1

|ρ|

|H (iρ)|B(X) (

1

= −1

≤ c

∞

2π 1

1

|S2 (t)|B(X) dt =

∞

2π 1

|H(is) − H(−is)|B(X) ds/s

0 1

|H(is) − H(0)|B(X) ds/|s| < ∞.

1 0

(|H (−is)|B(X) + |H (is)|B(X) )ds

1

|H (is)|B(X) ds < ∞.

= −1

∞

2π 1

|S4 (t)|B(X) dt ≤

1

(

0

≤ 2M

|ρ|≥s 1

|dH (iρ)|B(X) )ds

(

0

∞

1 −1

3

1

dρ/ρ )ds + 1

= M+

| sin τ |dτ /τ )dρ

|ρ|/2

1

−1

|S3 (t)|B(X) dt ≤

1/2

|H (iρ)|B(X) dρ < ∞.

−1

≤

| sin(|ρ|/2s)|ds/s)|H (iρ)|B(X) dρ

−1

(

|ρ| 0

ds)|dH (iρ)|B(X)

|ρ||dH (iρ)|B(X) < ∞.

Therefore S ∈ L1 ([1, ∞); B(X)), and since S(t) is also bounded in B(X) we see that S(t) is uniformly integrable in X. The second assertion follows by repeating the above arguments in the space Y . 2

280


Let us discuss (10.38) and (10.39) for equations of variational type t (w, u(t)) + α(t − s, w, u(s))ds = < w, f (t) >, t ≥ 0, w ∈ V,

(10.40)

0

d

where V → H are Hilbert spaces, and α : ⺢+ × V × V → ⺓ is a uniformly measurable and locally integrable bounded sesquilinear form on V satisfiying (PV1), ∗ ˆ (PV2), (PV3) with k = 2; see Section 7.3. Then A(λ) ∈ B(V, V ) is invertible and by (7.22) we have −1 ˆ |(λA(λ)) |B(V ∗ ,V ) ≤

1 , ¯ γϕ(λ)|λ|

Re λ > 0.

(10.41) ∗

¯ → ∞ as λ → 0; then by (10.41), F (λ) → 0 in B(V , V ) as λ → 0, Suppose |λ|ϕ(λ) i.e. (10.38) holds with F (0) = 0. From

ˆ F (λ) = −F (λ)(λA(λ)) F (λ)

and

ˆ ˆ ˆ F

(λ) = 2F (λ)(λA(λ)) F (λ)(λA(λ)) F (λ) − F (λ)(λA(λ)) F (λ)

as well as (PV4) with k = 2, we then obtain |F (λ)|B(V ∗ ,V ) ≤ C(

1 C ¯ ≤ )2 ϕ(λ) 2 ¯ ¯ , γϕ(λ)|λ| |λ| ϕ(λ)

Re λ > 0,

and similarly |F

(λ)|B(V ∗ ,V ) ≤

C ¯ , |λ|3 ϕ(λ)

Re λ > 0.

Therefore, if ϕ(λ) η admits continuous extension to ⺓+ \ {0}, (10.39) is implied by the condition −η (|ρ|2 ϕ(iρ))−1 dρ < ∞, for some η > 0. Corollary 10.7 Let the assumptions of Theorem 7.3 (ii) with k = 2 be satisfied and assume ϕ(λ) admits continuous extension to ⺓+ \ {0} such that η dρ ¯ = ∞ and lim |λ|ϕ(λ) < ∞ for some η > 0. (10.42) 2 λ→0 −η |ρ| ϕ(iρ) ∗

Then the resolvent S(t) for (10.40) is uniformly integrable in V , V and H. ¯ → ϕ(0) = 0 as λ → 0 and On the other hand, suppose |λ|ϕ(λ) ¯ u, v) →λ→0 α∞ (u, v) = < u, A∞ v > λˆ α(λ,

uniformly for ||u||, ||v|| ≤ 1.

∗

ˆ Then λA(λ) → A∞ in B(V, V ), where A∞ is invertible. Therefore F (λ) = ∗ −1 ˆ (λA(λ)) → A−1 ∞ in B(V , V ) as λ → 0, and so (10.38) follows. Since F (λ) ∗ is bounded in B(V , V ) it is easy to see that (10.39) is implied by |λˆ α( u, v) − α∞ (u, v)| + |λ(λα) ˆ (λ, u, v)| + |λ2 (λˆ α)

(λ, u, v)| ≤ ψ(λ)||u|| ||v|| (10.43)


281

for Re ηλ > 0, u, v ∈ V where ψ(λ) admits continuous extension to ⺓+ \ {0} such that −η ψ(iρ)dρ/|ρ| < ∞. Corollary 10.8 Let the assumptions of Theorem 7.3 (ii) with k = 2 be satis¯ → ϕ(0) = 0 and λˆ fied and suppose |λ|ϕ(λ) α(λ, u, v) → α∞ (u, v) uniformly for ||u||, ||v|| ≤ 1 as λ → 0, and let (10.43) hold, with a function ψ ∈ C(⺓+ \ {0}) such η that −η ψ(iρ)dρ/|ρ| < ∞ for some η > 0. Then the resolvent S(t) for (10.40) is ∗

uniformly integrable in V , V and H. 10.6 Comments a) L2 -stability has been considered in infinite dimensions for the first time by Miller and Wheeler [246]; although their definition is slightly different from ours the underlying ideas are similar. To be more precise, as regards to stability, Miller and Wheeler study the second order problem t x ¨(t) = cx(t) ˙ + (A + γ)x(t) + [b(t − τ )Ax(τ ) + g(t − τ )x(τ )]dτ + f (t), 0

x(0) = x0 , x(0) ˙ = v0 ,

t ≥ 0,

(10.44)

in a Hilbert space X, where A is a selfadjoint, negative definite operator in X such that (μ−A)−1 is compact for each μ ∈ ρ(A), c, γ ∈ ⺢, and b, g ∈ C 1 (⺢+ )∩L1 (⺢+ ). (10.44) is then called L2 -stable if it admits a mild solution x ∈ L2 (⺢+ ; X) for each x0 , v0 ∈ X and f ∈ L1 (⺢+ ; X). It is then proved that the spectral conditions 1 + ˆb(λ) = 0, (λ2 − cλ − γ − gˆ(λ))/(1 + ˆb(λ)) ∈ ρ(A), for all Re λ ≥ 0, (10.45) imply L2 -stability, provided in addition c + b(0) < 0 and b˙ ∈ L1 (⺢+ ). By means of the subordination principle and perturbation techniques which are discussed in Section 12.1, it can be shown that a stronger version of this result is true in a general Banach space for operators A generating a strongly continuous cosine family, g ∈ L1 (⺢+ ), and b ∈ W 1,1 (⺢+ ), provided the spectral condition (10.45) holds, and ω0 (A) < −(c + b(0)). b) The concept of local analyticity is originally due to Gelfand, Raikov, and Shilov [128], Paragraph 13, and is given in the context of a general commutative Banach algebra. The abstract version of Lemma 10.1 and a proof can be found there. This result is a considerable extension of the classical Wiener-Lévy theorem, and is a very powerful tool in modern transform theory; the results in Sections 10.2 and 10.3 are examples of this. The definition of a locally analytic function given here is a slight but nevertheless quite useful extension of the original concept, due to Jordan, Staffans, and Wheeler [189]. They study Volterra equations in finite dimensions in weighted spaces on the halfline and on the line. Based on their Proposition 2.3 (the result corresponding to Lemma 10.1) they were able to unify most results on integrability of resolvents available at that time, and to extend and strengthen many of them. In this connection we refer also to the monograph byGripenberg, Londen, and Staffans [156].

282


c) The first result on integrability of resolvents for parabolic problems appears already in the paper of Friedman and Shinbrot [123]. In the second part of this work it is proved that the resolvent S(t) of the equation of scalar type (10.1) belongs to Lp (⺢+ ; B(X)), p ≥ 2, if the kernel a satisfies 1,1 a ∈ Wloc (⺢+), a(0) > 0, a˙ ∈ L1 (⺢+ ),

the closed, linear, and densely defined operator A generates an analytic C0 -semigroup in X which is bounded in some sector Σ(0, θ), A is invertible, and the spectral condition a ˆ(λ) = 0, 1/ˆ a(λ) ∈ ρ(A),

for all Re λ ≥ 0,

holds. If in addition ta(t) ˙ is integrable then the uniform integrability of the resolvent is obtained. Obviously, the assumptions on a and A stated above imply Conditions (I1), (I2), (I3) of Theorem 10.1, however, in general the kernel need not be 2-regular, and therefore neither Theorem 10.2 nor Theorem 10.1 can be applied. But by means of parabolic perturbation results, we show in Section 12 that the moment condition on a(t) ˙ is not needed for uniform integrability of S(t). This was observed first in Pr¨ uss [271]; see also [275]. Theorems 10.1, 10.2, 10.7 are new. d) Theorems 10.3 and 10.4 on integrability of subordinate resolvents are unifications and extensions of results for the special cases b(t) = 1, and b(t) = t, i.e. the first and second order case, obtained in Pr¨ uss [274]. These special cases are stated here as Corollaries 10.2 and 10.3. By means of the methods presented in Section 10.3 it is also possible to discuss integrability of operator families related to S(t), ˙ ¨ like S(t), S(t), 1 ∗ S(t), and so on. For the second order case this has been done for the first two derivatives of S(t) in Pr¨ uss [277], and for (1 + t)S(t), 1 ∗ S(t), and ˙ (1 + t)2 S(t) even with a weight ρ in Hannsgen and Wheeler [171]. e) In analogy to Theorems 10.3 and 10.4 one obtains a characterization of integrability of the integral resolvent R(t) of subordinated equations. If the assumptions of Theorems 10.3 or 10.4 hold, R(t) is integrable iff the following conditions are satisfied. (IR1) 1/ˆ a(λ) admits a continuous extension in ⺓ to ⺓+ , holomorphic on ⺓+ ; (IR2) 1/ˆ a(λ) ∈ ρ(A) for all λ ∈ ⺓+ ; (IR3) 1/ˆ a(λ) is locally analytic on i ⺢. Note that these conditions differ from the corresponding ones for S(t) only at λ = 0. f) The papers Hannsgen [164], and Carr and Hannsgen [38], contain the first integrability results for resolvents S(t) of equations of scalar type (10.1) in a Hilbert space with negative definite operator A, where the kernels a(t) are -in our terminology- creep functions with a1 (t) convex. Assuming in addition a1 = a2 +a3 , ∞ where a2 is 3-monotone and a3 2-monotone and such that 1 a3 (t)dt/t < ∞, in


283

these papers it has been shown that the scalar functions s(t; μ) (cp. Section 1.3) are integrable, uniformly w.r.t. μ ≥ μ0 > 0, i.e. ∞ sup |s(t; μ)|dt < ∞, for each μ0 > 0. 0

μ≥μ0

(In case a(t) is even a Bernstein function Hannsgen and Wheeler [169] gave later a different argument which is based on the decomposition of s(t; μ) into a completely monotonic part which is infinitely smooth at t = 0, and an exponentially decreasing part.) Via the functional calculus for selfadjoint operators such estimates imply integrability of S(t), rather than strong integrability, i.e. as proved in Section 10.4; see Corollary 10.3. Although the analysis is beautiful, these estimates are of a very technical nature and therefore are not reproduced here. Later they have been extended by Carr and Hannsgen [39] and Noren [260] to obtain also integrability of ˙ ¨ S(t) and S(t). These estimates do not seem to carry over to nonscalar equations, however, as we have shown in Corollary 10.2, strong integrability of the resolvent is enough to obtain Lp -stability. Theorem 10.6 can therefore be considered as the natural extension of the Carr-Hannsgen result to nonscalar problems; cp. Pr¨ uss [281]. This paper contains also the straightforward application of Theorem 10.6 and its corollary to nonisotropic hyperbolic viscoelasticity. g) A topic very much related to stability of (10.1) and integrability of its associated resolvent S(t) is the question of stabilizability of (10.1) via feedback controls. For this subject the reader is referred e.g. to the papers Da Prato and Lunardi [74], Desch and Miller [89], [90], Jeong [188], [187], and Leugering [209], [211], [212].

11

Limiting Equations

The solvability behaviour of Volterra equations on the line will be studied in this section. The central concept which corresponds to well-posedness is that of admissibility of homogeneous spaces of functions on the line. Necessary conditions for admissibility are derived and some consequences of this property are studied. Thereafter the connections between the equations on the line and on the halfline are studied and the notion of limiting equation is justified. 11.1 Homogeneous Spaces d

Let X, Y be Banach spaces such that Y → X, {A(t)}t≥0 ⊂ B(Y, X) measurable, and consider the Volterra equation on the line ∞ u(t) = g(t) + A(τ )u(t − τ )dτ, t ∈ ⺢, (11.1) 0

L1loc (⺢; X).

where g ∈ This section is devoted to a study of the solvability behaviour of (11.1) in various spaces of functions on the line. In general, (11.1) is more difficult to handle than its local version (6.1), however, in contrast to (6.1) it enjoys the property of translation invariance, i.e. if u is a solution of (11.1) in the sense of Definition 11.2 below, then (Th u)(t) = u(t + h), t ∈ ⺢, is again a solution of (11.1), with g replaced by Th g, for each h ∈ ⺢. To be able to exploit this property, we consider the solvability behaviour of (11.1) only in spaces of locally integrable functions on ⺢ which are translation invariant, more precisely, homogeneous in the sense of Definition 11.1 A Banach space H(X) of locally integrable X-valued functions is said to be homogeneous if (i) H(X) → L1loc (⺢; X), i.e. for each T > 0 there is a constant C(T ) > 0 such that T

−T

|f (t)|dt ≤ C(T )|f |H(X)

for each f ∈ H(X);

(ii) {Th : h ∈ ⺢} ⊂ B(H(X)) is strongly continuous and uniformly bounded; (iii) B(X) → B(H(X)), i.e. f ∈ H(X), K ∈ B(X) imply Kf ∈ H(X); are satisfied. Examples of homogeneous spaces are Lp (X) := Lp (⺢; X), 1 ≤ p < ∞, the space of bounded uniformly continuous X-valued functions Cub (X) := Cub (⺢; X), the space of almost periodic functions AP (X), the space of continuous ω-periodic functions Pω (X), etc. L∞ (X) := L∞ (⺢; X) and Cb (X) := Cb (⺢; X) are not homogeneous since the group of translations is not strongly continuous in these spaces. The largest space we will consider is BM (X), the space of functions bounded in mean, defined by BM (X) = {f ∈ L1loc (⺢; X) : |f |b1 < ∞}, J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_11, © Springer Basel 1993

284

11. Limiting Equations

285

where the norm | · |b1 in BM (X) is given by t+1 b |f |1 = sup{ |f (s)|ds : t ∈ ⺢}. t

BM (X) is not homogeneous, however, its closed subspace BM 0 (X), defined by BM 0 (X) = {f ∈ BM (X) : |Th f − f |b1 → 0

as h → 0},

has this property; in fact, it is the largest homogeneous space, as (a) of the next proposition shows. If H(X) is homogeneous, the generator of the translation group will be denoted by DH , or simply by D = d/dt if there is no danger of confusion. We then may define m m Hm (X) = D(DH ), |f |Hm (X) := |Dl f |H(X) , m ∈ ⺞, l=0

and

H∞ (X) = ∩m≥1 Hm (X);

obviously Hm (X) are again Banach spaces, and H∞ (X) is a Fréchet space with the seminorms pm (f ) = |Dm f |H(X) , m ∈ ⺞0 . Of interest also is the space Ha (X) of entire vectors w.r.t. the translation group, i.e. 1/n

Ha (X) = {f ∈ H∞ (X) : limn→∞ |Dn f |H(X) < ∞}. This is precisely the space of all functions f ∈ H(X) such that Th f extends to an entire function on ⺓ of exponential growth. Since {Th } is a C0 -group on H(X) it is well-known that Ha (X) is dense in H(X); this is the second assertion of the next proposition. Proposition 11.1 Let H(X) be homogeneous. Then (i) H(X) → BM 0 (X); (ii) Ha (X) is dense in H(X); (iii) K ∈ BV (⺢; B(X)) induces via T f := dK ∗ f an operator there is a constant C > 0 such that |T | ≤ CVar K|∞ −∞ .

T ∈ B(H(X)),

Proof: (i) For f ∈ H(X) we have by (i) and (ii) of Definition 11.1 t+1 1 |f (s)|ds = |(Tt f )(s)|ds ≤ C(1)|Tt f |H(X) ≤ C(1)M |f |H(X) , 0

t

where M = sup{|Tt |B(H(X)) : t ∈ ⺢}. (ii) is standard in semigroup theory; see e.g. Davies [77], p. 30. (iii) follows from the representation ∞ ∞ T f = dK ∗ f = dK(τ )f (· − τ ) = dK(τ )T−τ f −∞

=

lim

δ→0

N i=1

(K(τi ) − K(τi−1 ))T−τi f,

−∞

t ∈ ⺢,

286


where δ = sup{τi − τi−1 : i = 1, . . . , N } denotes the size of the decomposition −∞ < τ0 < . . . < τN < ∞; observe that the function τ → Tτ f ∈ H(X) is uniformly continuous on ⺢. 2

In particular, any integrable function F : ⺢ → B(X) induces by Proposition 11.1 (iii) via T f := F ∗ f an operator T ∈ B(H(X)), for every homogeneous space H(X). Functions f ∈ BM (X) are of polynomial growth, since

1 0 ∞ dt 1 |f (t)| ≤ ( |Tj f (t)|dt + |T−j f (t)|dt) ≤ c|f |b1 , 2 1 + t2 1 + j −∞ 0 −1 j=0 ∞

for some constant c > 0, independent of f . Therefore, σ(f ), the spectrum of f , introduced in Section 0.5 via the Fourier-Carleman transform is well-defined. This gives rise to the family HΛ (X) of subspaces of a homogeneous space H(X), defined by HΛ (X) = {f ∈ H(X) : σ(f ) ⊂ Λ}. Theorem 0.8 shows that HΛ (X) is closed, whenever Λ ⊂ ⺢ is closed. For the spaces CubΛ (X) we use the abbreviation Λ(X), i.e. Λ(X) = {f ∈ Cub (X) : σ(f ) ⊂ Λ}. If H(X) is homogeneous and Λ ⊂ ⺢ is closed, then Proposition 0.4 implies that HΛ (X) is again homogeneous. This gives a fine structure of subspaces suitable to describe the solvability behaviour of (11.1) in terms of the spectrum of the inhomogeneity g. If H(X) is homogeneous, it is not difficult to verify Ha (X) ⊂ Hσ (X), where Hσ (X) = {f ∈ H(X) : σ(f ) compact}; therefore the space Hσ (X) is also dense in H(X). Next given b ∈ W(⺢) = {b ∈ L1 (⺢) : ˜b(ρ) = 0 for all ρ ∈ ⺢} ∪ {δ0 }, the Wiener class, we define Hb (X) = {f ∈ H(X) : f = b ∗ g

for some g ∈ H(X)},

i.e. Hb (X) is the range of the convolution operator on H(X) induced by b. By local analyticity of 1/˜b on ⺢ and by Proposition 0.6, there follows Hσ (X) ⊂ Hb (X), for each b ∈ W(⺢), hence these spaces are also dense in H(X). It should be noted that for the kernels bβ (t) = e0 (t)e−t tβ−1 /Γ(β), one obtains Hbβ (X) = Hβ (X) = D(Dβ ), for all β > 0; in fact, it is easy to verify the relation (I + D)−β f = bβ ∗ f . Let us summarize this discussion in Proposition 11.2 Let Λ ⊂ ⺢ be closed, H(X) be homogeneous, b ∈ W(⺢), and let HΛ (X), Ha (X), Hσ (X), Hb (X) be defined as above. Then (i) HΛ (X) is a homogeneous subspace of H(X);


287

(ii) Ha (X) ⊂ Hσ (X) ⊂ Hb (X); (iii) Hσ (X) and Hb (X) are dense in H(X). 11.2 Admissibility Let’s turn attention to (11.1) now. If A ∈ L1 (⺢+ ; B(Y, X)) and u ∈ BM 0 (Y ) then A ∗ u is well-defined, belongs to BM 0 (X), and |A ∗ u|b1 ≤ |A|1 |u|b1 . For many applications, however, integrability of A(t) is too stringent, but then the convolution A ∗ u over ⺢ is in general not defined. For this reason we restrict attention to kernels A(t) of the form A(t) = A0 (t) + A1 (t), A0 ∈ L1 (⺢+ ; B(Y, X))

t ≥ 0, where

and consider the differentiated version of (11.1): ∞ ∞ u(t) ˙ = A0 (τ )u(t ˙ − τ )dτ + dA1 (τ )u(t − τ ) + f (t), 0

0

(11.2)

and A1 ∈ BV (⺢+ ; B(Y, X)),

t ∈ ⺢.

(11.3)

In the sequel we always assume (11.2) and confine our study to (11.3). Remarks on (11.1) with A ∈ L1 (⺢+ ; B(Y, X)) and on the second order case where (11.1) is differentiated twice will be given in Section 11.7. Definition 11.2 Let f ∈ BM 0 (X). Then (a) u ∈ BM 0 (X) is called strong solution of (11.3) if u ∈ BM 0 (Y ), u−A0 ∗ u ∈ BM 1 (X), and (u − A0 ∗ u)· = dA1 ∗ u + f a.e. on ⺢. (b) u ∈ BM 0 (X) is called mild solution of (11.3) if there are fn ∈ BM 0 (X), fn → f in BM 0 (X), and strong solutions un ∈ BM 0 (Y ) of (11.3) with f replaced by fn , such that un → u in BM 0 (X). The solvability behaviour of (11.3) is described in terms of admissibility of homogeneous spaces which we introduce next. Definition 11.3 Let b ∈ W(⺢) be given. A homogeneous space H(X) is called b-admissible for equation (11.3), if for each f ∈ Hb (X) there is a unique strong solution u ∈ H(X) ∩ BM 0 (Y ) of (11.3), and (fn ) ⊂ Hb (X), fn → 0 in H(X), implies un → 0 in H(X). H(X) is called 0-admissible if it is δ0 -admissible. Thus the kernel b ∈ W(⺢) measures the amount of regularity in time of an inhomogeneity f needed to allow for a strong solution. Suppose the homogeneous space H(X) is b-admissible. Then we may define the solution operator G : Hb (X) → H(X) ∩ BM 0 (Y ) by means of Gf = u, where u ∈ H(X) ∩ BM 0 (Y ) denotes the unique solution of (11.3) with f ∈ Hb (X). By definition of b-admissibility, G admits a unique bounded linear extension to G ∈ B(H(X)) since Hb (X) is dense in H(X) by Proposition 11.1. Moreover, the closed graph theorem implies also b ∗ G ∈ B(H(X), BM 0 (Y )), where we deliberately use the notation b ∗ G for the operator G(b ∗ f ). Therefore, Gf is a mild solution of (11.3), for each f ∈ H(X).

288


Translation invariance and uniqueness of strong solutions for f ∈ Hb (X) then imply that G commutes with the group of translations, hence also with its generator D, in particular G ∈ B(Hm (X)) for each m ≥ 0, as well as b ∗ G ∈ B(Hm (X), BM m (Y )). Observe also the relation G(ϕ ∗ f ) = ϕ ∗ Gf,

f ∈ H(X), ϕ ∈ L1 (⺢),

(11.4)

which follows again from translation invariance and Proposition 11.1. These observations are collected in Proposition 11.3 Suppose the homogeneous space H(X) is b-admissible, and let the solution operator G be defined as above. Then (i) G ∈ B(Hm (X)), and b ∗ G ∈ B(Hm (X), BM m (Y )) for each m ≥ 0; (ii) G commutes with the translation group, i.e. Th G = GTh for all h ∈ ⺢; (iii) G commutes with D; (iv) G(ϕ ∗ f ) = ϕ ∗ Gf for all f ∈ H(X), ϕ ∈ L1 (⺢). Next we draw a connection between integrability properties of the resolvent S(t) for the local version (6.1) of (11.1) and admissibility. Proposition 11.4 Suppose (6.1) admits a resolvent S(t), which is integrable in B(X) and such that b ∗ S is integrable in B(X, Y ), for some b ∈ W(⺢+ ). Then each homogeneous space H(X) is b-admissible. Proof: We will show the representation ∞ (Gf )(t) = S(τ )f (t − τ )dτ, 0

t ∈ ⺢.

(11.5)

Since S(t) is integrable in B(X) by assumption, Proposition 11.1 shows G ∈ B(H(X)) for each homogeneous space. On the other hand, since b ∗ S is integrable in B(X, Y ) we obtain b ∗ G ∈ B(H(X), BM 0 (Y )). From the first resolvent equation (6.2) we obtain with R = b ∗ S (R − A0 ∗ R)· = b + dA1 ∗ R,

t ∈ ⺢,

hence u = b ∗ Gg = Gb ∗ g = R ∗ g with g ∈ H(X) is a strong solution of (11.3) for f = b ∗ g. Conversely, if u is a strong solution of (11.3), by the second resolvent equation (6.3), (S − S ∗ A0 )· = δ0 + S ∗ dA1 , we obtain u = S ∗ f , hence uniqueness. 2

To obtain conditions necessary for admissibility, suppose Λ(X) is b-admissible. For ρ ∈ Λ, the function f (t) = eiρt x, x ∈ X, belongs to Λa (X), hence there is a unique strong solution u ∈ Λ(X) ∩ BM 0 (Y ) of (11.3). Translation invariance, i.e. Proposition 11.3 then yields u˙ = DGf = Gf˙ = iρGf = iρu,


289

hence u(t) = eiρt y for some unique y ∈ Y . (11.3) now implies 1 (iρ))y = x, (iρ − iρAˆ0 (iρ) − dA 1 (iρ) = iρ(I − A(iρ)) ˆ hence the operators iρ − Aˆ0 (iρ) − dA ∈ B(Y, X) are invertible for each ρ ∈ Λ. Moreover, boundedness of G in Λ(X), and of b ∗ G from Λ(X) to BM 0 (Y ) yield the estimates |y|X = |u|Λ(X) ≤ |G|B(Λ(X)) |f |Λ(X) ≤ c|x|X , |˜b(ρ)||y|Y = |b ∗ u|BM 0 (Y ) ≤ |b ∗ G|B(Λ(X),BM 0 (Y )) |f |Λ(X) ≤ c|x|X . With the notation H(λ) =

1 −1 1 (λ))−1 , ˆ (I − A(λ)) = (λ − λAˆ0 (λ) − dA λ

(11.6)

which was used before, this implies |H(iρ)|B(X) + |˜b(ρ)||H(iρ)|B(X,Y ) ≤ M,

ρ ∈ Λ.

Defining the real spectrum of (11.3) by 1 (iρ) ∈ B(Y, X) is not invertible}, Λ0 = {ρ ∈ ⺢ : iρ − iρAˆ0 (iρ) − dA

(11.7)

we have proved Proposition 11.5 Suppose Λ(X) is b-admissible, Λ ⊂ ⺢ closed. Then Λ∩Λ0 = ∅, and there is a constant M > 0 such that |H(iρ)|B(X) + |˜b(ρ)||H(iρ)|B(X,Y ) ≤ M,

for all ρ ∈ Λ,

(11.8)

where H(λ) is defined by (11.6). The converse of Proposition 11.5 is in general false, even in the semigroup case A(t) ≡ A; see Section 12.7 for further remarks in the semigroup case. However, for large classes of equations or closed sets Λ ⊂ ⺢ characterization of admissibility in terms of the spectral condition Λ∩Λ0 = ∅ and uniform bounds on H(iρ) like (11.8) is possible. The next subsection and all of Section 12 deal with this question. 11.3 Λ-Kernels for Compact Λ A main tool in the sequel will be the following Lemma 11.1 Suppose ϕ ∈ L1 (⺢) is such that ϕ˜ ∈ C0∞ (⺢) and supp ϕ˜ ∩ Λ0 = ∅. Then there is Gϕ ∈ W ∞,1 (⺢; B(X, Y )) such that G˙ ϕ = A0 ∗ G˙ ϕ + dA1 ∗ Gϕ + ϕ = G˙ ϕ ∗ A0 + Gϕ ∗ dA1 + ϕ.

(11.9)

Moreover, Gϕ admits extension to an entire function Gϕ (z) with values in B(X, Y ) of exponential growth.

290


Proof: Since ϕ˜ has compact support and supp ϕ˜ ∩ Λ0 = ∅, H(iρ) is bounded in B(X, Y ) on supp ϕ, ˜ and therefore Gϕ (z) defined by ∞ 1 iρz H(iρ)ϕ(ρ)e ˜ dρ, z ∈ ⺓, (11.10) Gϕ (z) = 2π −∞ is an entire function with values in B(X, Y ). Clearly Gϕ and all its derivatives are bounded on ⺢ and of exponential growth on ⺓, and it is easily verified that (n) (11.9) holds. Thus it remains to prove Gϕ ∈ L1 (⺢; B(X, Y )) for each n ∈ ⺞0 . n (n) Replacing ϕ˜ by ϕ˜ · (iρ) , i.e. ϕ by ϕ , it is enough to show Gϕ ∈ L1 (⺢; B(X, Y )) (n) since Gϕ = Gϕ(n) . (i) Suppose first that in addition to (11.2) we have ∞ ∞ 2 |A0 (t)|B(Y,X) (1 + t )dt < ∞, and |dA1 (t)|B(Y,X) (1 + t2 )dt < ∞. 0

0

ˆ Then iρA(iρ) is twice continuously differentiable on ⺢, hence Φ(ρ) = H(iρ)ϕ(ρ) ˜ is so as well. Integrating by parts twice, (11.10) becomes ∞ 1 Gϕ (t) = − Φ

(ρ)eiρt dρ, 2πt2 −∞ and so we obtain |Gϕ (t)|B(X,Y ) ≤ C(1 + t2 )−1 , i.e. Gϕ ∈ L1 (⺢; B(X, Y )). (ii) To remove the moment conditions, consider Am (t) given by Am (t) = A0m (t) + A1m (t), t > 0, where A0m (t) = A0 (t) for t ≤ m, A0m (t) = 0 for t > m, A1m (t) = A1 (t) for t ≤ m, A1m (t) = A1 (m) for t > m. ˆ Then λAˆm (λ) → λA(λ) in B(Y, X) as m → ∞, uniformly on compact subsets of ⺓+ , and A0m , dA1m have moments of any order. The relation ˆ ˆ iρ(I − Aˆm (iρ)) = iρ(I − A(iρ))(I − H(iρ)iρ(Aˆm (iρ) − A(iρ)))

(11.11)

therefore shows that Hm (iρ) = (iρ − iρAˆm (iρ))−1 exists on a neighborhood U of supp ϕ˜ and |Hm (iρ)|B(X,Y ) ≤ M < ∞ on supp ϕ, ˜ provided m is large enough. (11.11) then yields ˆ H(iρ)ϕ(ρ) ˜ = Hm (iρ)ϕ(ρ) ˜ + H(iρ)ϕ(ρ)iρ[ ˜ A(iρ) − Aˆm (iρ)]Hm (iρ)ϕ˜1 (ρ), (11.12) where ϕ˜1 ∈ C0∞ is such that ϕ˜1 = 1 on supp ϕ, ˜ 0 ≤ ϕ˜1 ≤ 1, and supp ϕ˜1 ⊂ U . By ˜ m = ϕH step (i) of this proof there are Gm , Gjm ∈ L1 (⺢; B(X, Y )) such that G ˜ m, 1−j ˜ Gjm = (iρ) ϕ˜1 Hm . Let K = (A0 − A0m ) ∗ G0m + (dA1 − dA1m ) ∗ G1m ,


291

which belongs to L1 (⺢; B(X)); then (11.12) becomes the convolution equation Gϕ = Gm + Gϕ ∗ K.

(11.13)

Since = I − iρ(A −A m )Hm ϕ˜1 = (iρ − iρA m (1 − ϕ˜1 ) − iρA ϕ˜1 )Hm I −K is invertible on ⺢ by the choice of ϕ˜1 , if m is large enough, by Theorem 0.6 there is a resolvent kernel R ∈ L1 (⺢; B(X)) for (11.13), and so the representation Gϕ = Gm + Gm ∗ R shows Gϕ ∈ L1 (⺢; B(X, Y )). The proof is complete. 2

By means of Lemma 11.1 we can now prove the converse of Proposition 11.5 for compact Λ. Theorem 11.1 Suppose Λ ⊂ ⺢ is compact, Λ ∩ Λ0 = ∅, and let H(X) be homogeneous. Then HΛ (X) is 0-admissible and there is GΛ ∈ W ∞,1 (⺢; B(X, Y )) such that the solution operator G for HΛ (X) is represented by ∞ (Gf )(t) = GΛ (τ )f (t − τ )dτ, t ∈ ⺢, f ∈ HΛ (X). (11.14) −∞

GΛ (t) extends to an entire B(X, Y )-valued function of exponential growth. Proof: Since Λ ⊂ ⺢ is compact and Λ ∩ Λ0 = ∅, there is a neighborhood Λ3ε = Λ + B3ε (0) of Λ such that Λ3ε ∩ Λ0 = ∅; observe that Λ0 is closed. Choose ϕ0 ∈ L1 (⺢) such that ϕ˜0 ∈ C0∞ (⺢), ϕ˜0 ≡ 1 on Λε , ϕ˜0 ≡ 0 on ⺢ \ Λ2ε , 0 ≤ ϕ˜0 ≤ 1, and let GΛ ∈ W ∞,1 (⺢; B(X, Y )) denote the function Gϕ from Lemma 11.1 with ϕ = ϕ0 . If f ∈ HΛ (X), then σ(f ) ⊂ Λ hence ϕ0 ∗ f = f by Proposition 0.6, and so u = GΛ ∗ f is a strong solution of (11.3), by (11.9). On the other hand, if u ∈ HΛ (X) is a strong solution of (11.3) then (11.9) yields GΛ ∗ f = ϕ0 ∗ u = u by Proposition 0.6 again, hence uniqueness. Since GΛ ∈ L1 (⺢; B(X, Y )), we finally obtain boundedness of the solution operator G in HΛ (X), as well as from HΛ (X) to BM 0 (Y ), and so HΛ (X) is 0-admissible. 2

A kernel GΛ representing the solution operator G of a homogeneous space HΛ (X) via (11.14) will be called a Λ-kernel in the sequel. Observe that 0-admissible implies b-admissible for any b ∈ W(⺢). Lemma 11.1 also yields dense solvability and uniqueness in HΛ (X) whenever Λ ∩ Λ0 = ∅, as we show next. Proposition 11.6 Let H(X) be homogeneous and suppose Λ ∩ Λ0 = ∅. Then (i) σ(u) ⊂ Λ0 for each strong solution of (11.3) with f = 0; (ii) for each f ∈ HΛ (X) there is at most one strong solution u ∈ HΛ (X);

292


σ (iii) for each f ∈ HΛ (X) there is a strong solution u ∈ HΛ (X); (iv) if f ∈ HΛ (X), and u ∈ HΛ (X) is a strong solution, then σ(u) = σ(f ).

˜ u , where Du denotes the distribution generated by Proof: (i) Since σ(u) = supp D u ∈ BM 0 (Y ) ⊂ BM 0 (X), in virtue of Proposition 0.5, we have to show ˜ u , ϕ] ˜˜ = (ϕ ∗ u)(0) = 0 [D ˜ = [Du , ϕ] for each ϕ˜ ∈ C0∞ (⺢) with supp ϕ˜ ∩ Λ0 = ∅. But given such ϕ, let Gϕ denote the kernel from Lemma 11.1; (11.9) and (11.3) then imply ϕ ∗ u = 0, in particular (ϕ ∗ u)(0) = 0. (ii) follows from (i), since σ(u) ⊂ Λ0 ∩ Λ = ∅ implies u = 0, by Proposition 0.5. σ (iii) If f ∈ HΛ (X) then σ(f ) ⊂ Λ is compact and does not intersect Λ0 . Applying Theorem 11.1 to the set σ(f ) we obtain a strong solution u ∈ Hσ(f ) (X) ⊂ HΛ (X) of (11.3). (iv) Suppose u ∈ HΛ (X) is a strong solution of (11.3), and let ϕ˜ ∈ C0∞ (⺢) be such that supp ϕ˜ ∩ σ(u) = ∅. Then σ(ϕ ∗ u) = ∅, hence ϕ ∗ u = 0 by Proposition 0.5, ˜ f , ϕ] and so (11.3) implies also ϕ ∗ f = 0. But this implies as above [D ˜ = 0, i.e. σ(f ) ⊂ σ(u). Conversely, let ϕ˜ ∈ C0∞ (⺢) satisfy supp ϕ˜ ∩ σ(f ) = ∅. Then ϕ ∗ f = 0, hence ϕ ∗ u is a strong solution of (11.3) with f = 0. Since σ(ϕ ∗ u) ⊂ supp ϕ, ˜ but also σ(ϕ ∗ u) ⊂ Λ0 by (i) of this proposition, we obtain σ(ϕ ∗ u) = ∅, hence ϕ ∗ u = 0 by Proposition 0.5 again; this implies σ(u) = σ(f ). 2

A reasoning similar to that of the last step in the proof of Proposition 11.6 yields Corollary 11.1 Suppose H(X) is a homogeneous space which is b-admissible, and let G denote the solution operator. Then σ(Gf ) = σ(f )

for each f ∈ H(X).

Proof: Consider first f ∈ Hb (X) and let u = Gf . Then as in (iv) of the proof of Proposition 11.6 we have σ(f ) ⊂ σ(u). Conversely, let ϕ˜ ∈ C0∞ (⺢) satisfy supp ϕ˜ ∩ σ(f ) = ∅, i.e. ϕ ∗ f = 0 by Proposition 0.6. Then ϕ ∗ u belongs to H(X) and is a strong solution of (11.3) with f = 0, hence by admissibility ϕ ∗ u = 0. With u , ϕ] [D ˜ = (ϕ ∗ u)(0) this then implies σ(u) = σ(f ), i.e. we have σ(Gf ) = σ(f ) for each f ∈ Hb (X). For the general case consider ϕ˜ ∈ C0∞ (⺢) such that 0 ≤ ϕ˜ ≤ 1, ϕ(ρ) ˜ = 1 for |ρ| ≤ 1, ϕ(ρ) ˜ = 0 for |ρ| ≥ 2, and let ϕn (t) = nϕ(nt). ϕn is then an approximation of the identity, i.e. ∞ ∞ ϕn ∗ f = ϕn (τ )T−τ f dτ = ϕ(τ )T−τ /n f dτ → f in H(X) −∞

−∞

since the translation group is strongly continuous in H(X). But σ(ϕn ∗f ) ⊂ σ(f )∩ ¯2n (0) is compact, i.e. ϕn ∗ f ∈ Hσ (X). From Proposition supp ϕñ ⊂ σ(f ) ∩ B


293

11.3 we obtain ϕn ∗ Gf = Gϕn ∗ f , hence these are strong solutions, and so σ(ϕn ∗ Gf ) = σ(ϕn ∗ f ). But with Theorem 0.8 we obtain σ(Gf ) = ∩m≥1 ∪n≥m σ(ϕn ∗ Gf ) = ∩m≥1 ∪n≥m σ(ϕn ∗ f ) = σ(f ). 2

From Proposition 11.6 it is evident, that the main part in proving admissibility of HΛ (X), where Λ ∩ Λ0 = ∅, consists in establishing a bound M such that |u|H(X) ≤ M |f |H(X)

and |b ∗ u|BM 0 (Y ) ≤ M |f |H(X)

holds for each f ∈ HΛ (X). This question is studied for unbounded sets Λ ⊂ ⺢ in Section 12. Here we discuss next further consequences of admissibility. 11.4 Almost Periodic Solutions Recall that a function f ∈ Cb (X) is called almost periodic (a.p.) if {Tτ f : τ ∈ ⺢} is relatively compact in Cb (X). Since almost periodic functions are uniformly continuous, the space AP (X) of all a.p. functions is a closed translation invariant subspace of Cub (X), which is also homogeneous. Note that periodic functions are characterized among a.p. functions by the stronger property that {Tτ f : τ ∈ ⺢} is compact in Cb (X). The Bohr transform given by N α(ρ, f ) : = lim N −1 e−iρt f (t)dt, ρ ∈ ⺢, f ∈ AP (X), N →∞

0

is well-defined and continuous from AP (X) to X. For f ∈ AP (X) its exponent set exp(f ) = {ρ ∈ ⺢ : α(ρ, f ) = 0} is at most countable, and σ(f ) = exp(f ). This can be proved by means of Bochner’s approximation theorem, which states that, given a fixed countable subset {ρj }∞ 1 , there are convergence factors γnj ∈ ⺢ with γnj → 1 as n → ∞ such that the trigonometric polynomials Bn f =

n

γnj α(ρj , f )eiρj t ,

n ∈ ⺞,

j=1

converge to f uniformly on ⺢, provided exp(f ) ⊂ {ρj }∞ 1 . By virtue of this result, it is also clear that f ∈ AP (X) is uniquely determined by its Bohr transform. All of these results and many others can be found in the monograph of Amerio and Prouse [9]. Recall also that a function f ∈ Cub (X) is called asymptotically almost periodic (a.a.p.) (to the right) if {Tτ f |[0,∞) }τ ≥0 ⊂ Cb (⺢+ ; X) is relatively compact. It can be shown (see e.g. Fréchet [119]) that then there is an a.p. function fa such that f0 (t) = f (t) − fa (t) → 0 as t → ∞, i.e. f0 ∈ C0+ (X), where C0+ (X) = {f ∈ Cub (X) : f (t) → 0

as t → ∞}

294


is another closed subspace of Cub (X). The a.p. function fa is necessarily unique, since α(ρ, fa ) = α(ρ, f ), ρ ∈ ⺢, and the Bohr transform is unique. Moreover, |fa |∞ ≤ |f |∞ , and therefore we have the direct sum decomposition of the space AAP (X) of a.a.p. functions AAP + (X) = AP (X) ⊕ C0+ (X). The space

Cl+ (X) = {f ∈ Cub (X) : lim f (t) = f (∞) exists } t→∞

is a closed subspace of AAP (X). It is easy to see that C0+ (X), Cl+ (X), and AAP + (X) are homogeneous. After these preparations we are ready to prove the following result on a.p. and a.a.p. solutions of (11.3). +

Theorem 11.2 Suppose Λ(X) is b-admissible, where Λ ⊂ ⺢ is closed, and let G denote the corresponding solution operator. Then (i) f ∈ AP (X), exp(f ) ⊂ Λ, imply Gf ∈ AP (X), exp(Gf ) = exp(f ), and α(ρ, Gf ) = H(iρ)α(ρ, f ),

ρ ∈ exp(f ).

(11.15)

(ii) f ∈ AAP + (X), σ(f ) ⊂ Λ, imply Gf ∈ AAP + (X), (Gf )0 = Gf0 , (Gf )a = Gfa and also (11.15) holds; (iii) 0 ∈ Λ, f ∈ Cl+ (X), σ(f ) ⊂ Λ imply Gf ∈ Cl+ (X) and (Gf )(∞) = H(0)f (∞).

(11.16)

Proof: (i) Suppose Λ(X) is b-admissible. Then if ρ ∈ Λ, as in the discussion before Proposition 11.5 we obtain G(xeiρt ) = H(iρ)xeiρt ; note that Λ ∩ Λ0 = N iρn t ∅. Hence if f (t) = is a trigonometric polynomial with exp(f ) = n=1 fn e N N iρn t {ρn }1 ⊂ Λ we obtain by linearity (Gf )(t) = , i.e. (i) of n=1 H(iρn )fn e Theorem 11.2 holds for trigonometric polynomials. The general case then follows by Bochner’s approximation theorem mentioned above since G is bounded and the Bohr transform is continuous. (ii) If f = fa + f0 ∈ AAP + (X) ∩ Λ(X), then Tτn f → fa uniformly on compact subsets of ⺢, for some sequence τn → ∞. Hence by Theorem 0.8 we obtain σ(fa ) ⊂ σ(f ) ⊂ Λ, and so σ(f0 ) = σ(f − fa ) ⊂ σ(f ) ∪ σ(fa ) ⊂ σ(f ) as well. Since Λ(X) is b-admissible we therefore have Gf = Gfa + Gf0 , where Gfa ∈ AP (X) by step (i) of this proof, and (11.15) holds again. Thus it remains to show Gf0 ∈ C0+ (X), i.e. (Gf0 )(t) → 0 as t → ∞. By Proposition 11.2 it is enough to assume σ(f0 ) compact, i.e. w.l.o.g. Λ compact. Let GΛ ∈ W ∞,1 (⺢; B(X, Y )) denote a Λ-kernel for (11.3) which exists by Theorem 11.1. Then Gf0 = GΛ ∗ f0 and since GΛ ∈ L1 (⺢; B(X, Y )), (Gf0 )(t) → 0 as t → ∞ follows from f0 (t) → 0 as t → ∞. (iii) This is a consequence of (ii) with fa (t) ≡ f (∞). 2


295

Observe that in case f = b ∗ g, with g ∈ Λ(X) ∩ AP (X) and Λ(X) is b-admissible then u ∈ BM 0 (Y ) is a.p. in the sense of Stepanov in Y , i.e. {Tt u : t ∈ ⺢} ⊂ BM 0 (Y ) is relatively compact. If f ∈ Cub (X) is ω-periodic, then exp(f ) ⊂ (2π/ω)⺪ and 1 ω −2πint/ω α(2πn/ω, f ) = fn = e f (t)dt, n ∈ ⺪, ω 0 are the Fourier coefficients of f . Then with Λ = (2π/ω)⺪ we have Λ(X) = Pω (X), and Theorem 11.2 yields the following Corollary 11.2 Suppose Λ = (2π/ω)⺪ and Λ(X) is b-admissible. Then for every f ∈ Pω (X) there is a unique mild solution u ∈ Pω (X) of (11.3) and for their Fourier coefficients we have un = H(2πin/ω)fn ,

n ∈ ⺪.

(11.17)

In passing we note the following result for a.p. solutions, without assuming any admissibility. Proposition 11.7 Let f ∈ AP (X) and suppose u ∈ BM 0 (Y ) is a strong solution of (11.3) which is a.p. in Y in the sense of Stepanov. Then 1 (iρ))α(ρ, u) = α(ρ, f ) (iρ − iρAˆ0 (iρ) − dA

for all ρ ∈ ⺢.

(11.18)

In particular, the condition 1 (iρ)), α(ρ, f ) ∈ R(iρ − iρAˆ0 (iρ) − dA

ρ ∈ ⺢,

is necessary for existence of u. This follows from direct computation on the Fourier coefficients of u by using (11.3). Another concept of almost periodicity was introduced by Eberlein [107] and is defined as follows. A function f ∈ Cb (X) is called weakly almost periodic (w.a.p.) in the sense of Eberlein, if {Tτ f : τ ∈ ⺢} is weakly relatively compact in Cb (X). The space W (X) of all w.a.p. functions can be shown to be a closed translation invariant subspace of Cub (X) which is also homogeneous. The inclusion AP (X) ⊂ W (X) is obvious. According to the decomposition theorem of Jacobs, Glicksberg, Deleeuw (see e.g. Krengel [201], Sect. 2.4), W (X) = AP (X) ⊕ W0 (X), where the homogeneous space W0 (X) ⊂ W (X) is given by W0 (X) = {f ∈ W (X) :

w − lim Tτn f = w − lim Tσn f = 0, n→∞

n→∞

for some sequences τn → ∞, σn → −∞}.

296


Similarly, a function f ∈ Cub (X) is called weakly asymptotically almost periodic (w.a.a.p.) if {Tτ f |[0,∞) }τ ≥0 ⊂ Cb (⺢+ ; X) is weakly relatively compact. The space of all w.a.a.p. functions is also a closed translation invariant subspace of Cub (X) which will be denoted by W + (X). There is again a decomposition following from the Jacobs-Glicksberg-Deleeuw theorem. W + (X) = AP (X) ⊕ W0+ (X), where W0+ (X) is definded according to W0+ (X) = {f ∈ W + (X) : w − lim Tτn f |[0,∞) = 0 in Cb (⺢+ ; X), n→∞

for some sequence τn → ∞}; both spaces W + (X) and W0+ (X) are easily seen to be homogeneous and clearly W0 (X) ⊂ W0+ (X). For f ∈ W + (X) the Bohr transform of f is still well-defined and with f = fa + f0 , where fa ∈ AP (X), f0 ∈ W0+ (X), we have N −1 α(ρ, f ) = lim N e−iρt f (t)dt = α(ρ, fa ), ρ ∈ ⺢. (11.19) N →∞

0

This is implied by the general mean ergodic theorem for bounded C0 -semigroups; see e.g. Hille and Phillips [180], Chap. XVIII, and Ruess and Summers [298]. Furthermore, as in the case of AAP + (X) we have the relation σ(f ) = σ(fa ) ∪ σ(f0 ),

f ∈ W + (X).

(11.20)

In fact, the inclusion ⊂ is obvious by Proposition 0.4; the converse follows from Proposition 0.7 and (11.19), since ρ ∈ σ(f ) implies α(ρ, f ) = 0, hence α(ρ, fa ) = 0, which in turn yields ρ ∈ exp(fa ), i.e. σ(f ) ⊃ exp(fa ). Closedness of σ(f ) then gives σ(fa ) = exp(fa ) ⊂ σ(f ), but also σ(f0 ) = σ(f − fa ) ⊂ σ(f ) ∪ σ(fa ) ⊂ σ(f ), by Proposition 0.4. These arguments show that the second part of Theorem 11.2 remains valid if AAP + (X) is replaced by W + (X) and by W (X), thanks to Proposition 11.1 and to homogeneity of the spaces W0+ (X) and by W0 (X). Corollary 11.3 Suppose Λ(X) is b-admissible for some b ∈ W(⺢), where Λ ⊂ ⺢ is closed, and let G denote the corresponding solution operator. Then f ∈ W + (X) and σ(f ) ⊂ Λ imply Gf ∈ W + (X), (Gf )a = Gfa , (Gf )0 = Gf0 , and (11.15) holds. The same assertions are true for W (X) instead of W + (X). 11.5 Nonresonant Problems Consider (6.1) in differentiated form, i.e. t t u(t) ˙ = A1 (τ )u(t ˙ − τ )dτ + dA2 (τ )u(t − τ ) + f (t), 0

u(0) = u0 ,

0

t ∈ ⺢+ , (11.21)


297

as well as its limiting equation ∞ A1 (τ )v(t ˙ − τ )dτ + v(t) ˙ = 0

∞ 0

dA2 (τ )v(t − τ ) + f (t),

t ∈ ⺢.

(11.22)

The purpose of this and the next subsection is the study of the relations between the solutions of (11.21) and (11.22). For this we need the (complex) spectrum of (11.21). 2 (λ) ∈ B(Y, X) Σ0 = {λ ∈ ⺓+ : λ − λAˆ1 (λ) − dA

is not invertible}.

(11.23)

A standing assumption will be the well-posedness of (11.21), i.e. we take for granted the existence of an exponentially bounded weak resolvent S(t) for (11.21). Concerning (11.22) suppose the (real) spectrum Λ0 of (11.22) satisfies Λ0 = ∅ and there is an ⺢-kernel G(t) for (11.22); recall that G : ⺢ → B(X) satisfies at least (G1) G(·)x ∈ L1 (⺢+ ; X) for each x ∈ X; ˜ (G2) G(ρ) = H(iρ) for each ρ ∈ ⺢. Here as before we use the notation 2 (λ))−1 , H(λ) = (λ − λAˆ1 (λ) − dA

λ ∈ Σ0 .

(11.24)

To draw the connection between S(t) and G(t) we need in addition compactness of Σ0 and a uniform bound on H(λ) in B(X) away from Σ0 . Let Γ0 ⊂ ⺓+ denote some Jordan curve surrounding Σ0 counter-clockwise, and define 1 S0 (z) = eλz H(λ)dλ, z ∈ ⺓; (11.25) 2πi Γ0 clearly, S0 : ⺓ → B(X, Y ) is an entire function of exponential growth, and |S0 (t)|B(X,Y ) ≤ M eηt/2 ,

t ≤ 0,

(11.26)

where η = dist (Σ0 , i ⺢) = inf Re Σ0 . S0 satisfies the equations ∞ ∞ ˙ ˙ S0 (t) = A1 (τ )S0 (t − τ )dτ + dA2 (τ )S0 (t − τ ), t ∈ ⺢, 0

and

S˙ 0 (t) =

∞ 0

(11.27)

0

S˙ 0 (t − τ )A1 (τ )dτ +

∞ 0

S0 (t − τ )dA2 (τ ),

t ∈ ⺢,

(11.28)

as a simple computation shows. Let c(t) = e−t e0 (t), and define S(t) ≡ 0 for t < 0; then for ω > ω0 (S), the growth bound of S(t), the inversion formula (0.9) for the vector-valued Laplace transform yields ω+iN 1 (c ∗ S)(t) = lim cˆ(λ)H(λ)eλt dλ, t ∈ ⺢. (11.29) N →∞ 2πi ω−iN

298


Similarly, the inversion formula of the Fourier transform gives 1 (c ∗ G)(t) = lim N →∞ 2πi

N

cˆ(iρ)H(iρ)eiρt dρ,

−N

t ∈ ⺢;

(11.30)

(11.29) and (11.30) have to be understood in the strong sense in X. By means of Cauchy’s theorem we then obtain the identity (c ∗ S)(t) = (c ∗ S0 )(t) + (c ∗ G)(t),

t ∈ ⺢,

which after a differentation becomes S(t) = S0 (t) + G(t),

t ∈ ⺢.

(11.31)

From (11.31) further properties of the ⺢-kernel G(t) can be deduced. (G3) G(·)x is continuous on ⺢ \ {0}, for each x ∈ X; (G4) The limits G(0+) = limt→0+ G(t) and G(0−) = limt→0− G(t) exist strongly, and the jump relation G(0+) − G(0−) = I is satisfied. Moreover, G(t) also fulfills the second resolvent equation on the line, i.e.

t

G(t)y = −∞

G(τ )A(t − τ )ydτ + e0 (t)y,

t ∈ ⺢ \ {0}, y ∈ Y,

(11.32)

which follows from (G1), (G2) and (11.26), (11.31). Note that G(t) = −S0 (t) for t < 0, hence G(t) behaves very nicely for negative t. On the other hand, for t > 0, G(t) enjoys the same regularity properties as S(t), in particular if S(t) is a resolvent, the first resolvent equation on the line is valid too, i.e.

t

G(t)y = −∞

A(t − τ )G(τ )ydτ + e0 (t)y,

t ∈ ⺢ \ {0}, y ∈ Y.

(11.33)

However, this way it is not possible to obtain more information about the asymptotic behaviour of G(t), like boundedness as t → ∞, or limt→∞ G(t)x = 0 for each ˜ x ∈ X. Here the relation G(ρ) = H(iρ) must be employed. Let us summarize the above discussion. Theorem 11.3 Suppose A(t) is of the form (11.2), let Σ0 be defined by (11.23) and H(λ) by (11.24); assume in addition (i) Σ0 ∩ i ⺢ = ∅, Σ0 is compact and lim|λ|→∞ |H(λ)|B(X) < ∞;

(11.34)

(ii) there is a weak resolvent S(t) for (11.21) with growth bound ω0 (S) < ∞; set S(t) = 0 for t < 0; (iii) there is an ⺢-kernel G(t) for (11.22) satisfying (G1) and (G2).


299

Then S0 (t) = S(t) − G(t), t ∈ ⺢, belongs to C ∞ (⺢; B(X, Y )), and satisfies (11.26), (11.27) and (11.28); S0 (t) admits extension to an entire function of exponential growth. In particular, if Σ0 = ∅, then S(t) is strongly integrable. Observe that Theorem 11.3 yields further results on integrability of resolvents, whenever Σ0 = ∅, H(λ) is bounded on ⺓+ , once there is a weak resolvent for (11.21) of subexponential growth, and there is an ⺢-kernel for (11.22) satisfying (G1) and (G2). For the latter consult Section 12. A remark about compactness of Λ0 seems to be necessary. In hyperbolic situations one cannot expect that Λ0 is compact, in general, hence assumption (i) of Theorem 11.3 is a serious restriction. But even in the parabolic case there are examples showing that Λ0 need not be compact. Example 11.1 Let A0 be a selfadjoint negative definite unbounded linear operator in the Hilbert space X, Y = XA0 , and consider the problem u(t) ˙ = A0 u(t) + βA0 u(t − 1) + f (t),

t ∈ ⺢,

where β ∈ ⺢. This problem is clearly of the form (11.22) with A1 (t) ≡ 0 and A2 (t) = A0 e0 (t) + βA0 e0 (t − 1). We obtain by a simple computation Σ0 = {λ ∈ ⺓+ : λ/(1 + βe−λ ) ∈ σ(A0 ) ∪ {∞}}. For |β| < 1 we have 1 + βe−λ = 0 on ⺓+ and ϕ(λ) = λ(1 − βe−λ )−1 stays within the sector Σ(0, π/2 + arcsin |β|), hence Σ0 = ∅. But for β = −1 say, zeros of 1 − e−λ pop up at λ = 2πin, n ∈ ⺪, and so Λ0 = 2π⺪, in particular Σ0 is no longer compact. If β decreases below −1 then these zeros are shifted to the right by the amount log |β|. These are poles for ϕ(λ), hence they are limit points of sequences (λn ) ⊂ ⺓+ such that ϕ(λn ) ∈ σ(A0 ), i.e. the structure of Σ0 becomes quite complicated, although Λ0 can be even empty, and |H(iρ)| + |A0 H(iρ)|(1 + |ρ|)−1 bounded on ⺢. 2

Observe that the resolvent S(t) for Example 11.1 was shown to exists in Example 1.1. From the explicit formula given there one can see that S(t) is uniformly integrable provided |β| < 1. 11.6 Asymptotic Equivalence Suppose S(t) is a weak resolvent for (11.21) with finite growth bound ω0 (S), let Σ0 be compact, assume (11.34) holds, and let G(t) be an ⺢-kernel for (11.22) which satisfies (G1) and (G2), hence also (G3) and (G4) as shown in Section 11.5. Assume in addition that G(t) is also subject to (G5) lim|t|→∞ G(t)x = 0 for each x ∈ X, this is the case in all of those results in Section 12 claiming the existence of an ⺢-kernel. Last but not least let Cub (X) be b-admissible for some b ∈ W(⺢).

300


We want to consider the relations between the solutions u(t) and v(t) of (11.21) and (11.22). So let u0 ∈ X and f ∈ Cub (X); then u(t) = S(t)u0 + and

∞

v(t) = −∞

t 0

S(t − τ )f (τ )dτ,

G(t − τ )f (τ )dτ,

t ≥ 0,

t ∈ ⺢.

(11.35)

(11.36)

Subtracting these equations and using the decomposition (11.31) we obtain

∞

u(t) − v(t) = G(t)u0 −

G(τ )f (0)dτ −

t

where

w(t) = S0 (t)u0 +

∞ 0

0 −∞

G(t − τ )[f (τ ) − f (0)]dτ + w(t), (11.37)

S0 (t − τ )f (τ )dτ,

t ∈ ⺢.

(11.38)

∞ By (G5), G(t)u0 → 0, and t G(τ )f (0)dτ → 0 as t → ∞ by (G1); since C0+ (X) is also b-admissable there follows 0 G(t − τ )[f (τ ) − f (0)]dτ = Gf− (t) → 0 as t → ∞, −∞

with f− (t) = (f (t) − f (0))(1 − e0 (t)). Thus if we can show w(t) → 0 as t → ∞, then the solutions u(t) of (11.21) and v(t) of (11.22) are asymptotic to each other. Since S0 (t) is exponentially growing when nontrivial, it is not at all clear whether w ∈ C0 (⺢+ ; X), and in general this will not be true. However, if u(t) stays bounded as t → ∞, then it is, we prove below, even w(t) ≡ 0. This is the main result of this subsection, saying that for bounded solutions (11.21) and (11.22) are asymptotically equivalent. Theorem 11.4 Suppose A(t) is of the form (11.2), let Σ0 be defined by (11.23) and H(λ) by (11.24); assume in addition (i) Σ0 ∩ i ⺢ = ∅, Σ0 is compact, and lim|λ|→∞ |H(λ)|B(X) < ∞; (ii) there is a weak resolvent S(t) for (11.21) with growth bound ω0 (S) < ∞; (iii) the space Cub (X) is b-admissible for (11.22), for some b ∈ W(⺢); (iv) there is an ⺢-kernel G(t) for (11.22) satisfying (G1), (G2), (G5). Let u0 ∈ X, f ∈ Cub (X), and let u(t) and v(t) be given by (11.35) and (11.36); assume u(t) is bounded on ⺢+ . Then u(t) − v(t) → 0 as t → ∞. In particular, (a) f ∈ AAP + (X) implies u ∈ AAP + (X), and α(ρ, u) = H(iρ)α(ρ, f ),

ρ ∈ ⺢;

(11.39)


301

(b) f ∈ W + (X) implies u ∈ W + (X), and (11.39); (c) f ∈ Cl+ (X) implies u ∈ Cl+ (X), and u(∞) = H(0)f (∞). Proof: Assertions (a), (b) and (c) follow from Theorem 11.2 and Corollary 11.3, once u(t) − v(t) → 0 as t → ∞ has been proved. This in turn is a consequence of w(t) ≡ 0, as explained above. So let u(t) be bounded and let w(t) be given by (11.38). Then (11.37) shows that w(t) is also bounded, hence w(t) ≡ 0 if we can show σ(w) = ∅. The FourierCarleman transform w(λ) ˆ is defined for Re λ = 0; however, (11.26) and (11.38) yield |w(t)| ≤ M eηt (|u0 | + 2|f |∞ /η), t ≤ 0, hence w(λ) ˆ admits a holomorphic extension from the left halfplane Re λ < 0 to the halfplane Re λ < η. For 0 < Re λ < η, (11.38) leads to a representation for this extension, namely w(λ) ˆ = −

0

e

−λt

−∞

[S0 (t)u0 +

∞

(

∞ 0

S0 (t − τ )f (τ )dτ ]dt

0

S0 (t − τ )e−λ(t−τ ) dt)f (τ )e−λt dτ 0 −∞ ∞ τ = Sˆ0 (λ)u0 + Sˆ0 (λ)fˆ(λ) + ( S0 (t)e−λt dt)f (τ )dτ, = Sˆ0 (λ)u0 −

0

0

for all 0 < Re λ < η; observe that all integrals are absolutely convergent. On the other hand, for Re λ > ω = sup{ Re μ : μ ∈ Γ0 } (cp. (11.25)) we obtain by (11.38) ∞ ∞ w(λ) ˆ = e−λt [S0 (t)u0 + S0 (t − τ )f (τ )dτ ]dt 0 0 ∞ ∞ = Sˆ0 (λ)u0 + ( e−λ(t−τ ) S0 (t − τ )dτ )f (τ )e−λτ dt 0 0 ∞ τ ˆ ˆ ˆ = S0 (λ)u0 + S0 (λ)f (λ) + ( S0 (t − τ )e−λt dt)f (τ )dτ, 0

0

for all 0 < Re λ < η, where again all integrals are absolutely convergent. The last term in this representation is holomorphic in Re λ > 0, fˆ(λ) is so as the Laplace transform of a bounded function, and the Fourier Carleman transform Sˆ0 (λ) of S0 (t) is holomorphic outside the Jordan curve Γ0 ; in fact, 1 Sˆ0 (λ) = H(μ)dμ/(λ − μ), λ outside of Γ0 . 2πi Γ0 Therefore the extension of w(λ) ˆ to the halfplane Re λ < η coincides with w(λ) ˆ in the strip 0 < Re λ < η, i.e. w(λ) ˆ is entire. This implies σ(w) = 0, hence w(t) ≡ 0, by Proposition 0.5. 2

302


Let us briefly discuss the question when u(t) will be bounded. From the decomposition S = S0 + G and from (11.35) it is clear that u(t) is bounded iff t z(t) = S0 (t)u0 + S0 (t − τ )f (τ )dτ, t ≥ 0, (11.40) 0

is bounded. Therefore a necessary condition for boundedness of u(t) is that zˆ(λ) = Sˆ0 (λ)(u0 + fˆ(λ))

(11.41)

is holomorphic in the right halfplane, i.e. u0 + fˆ(λ) must ‘cancel’ the singularities of Sˆ0 (λ). In case Σ0 = {λ1 , . . . , λN } finite we obtain even necessary and sufficient conditions for boundedness of u(t). For this purpose consider the Laurent expansion of H(λ) at λ = λj , i.e. ∞

H(λ) =

Hj,m (λ − λj )−m + Hj (λ).

(11.42)

m=1

Then by residue calculus S0 (t) becomes S0 (t) =

N

eλ j t (

z(t) =

N j=1

where

∞

zj,l (t) =

m=l

Hj,m

m=1

j=1

hence

∞

eλj t

tm−1 ), (m − 1)!

∞ tl−1 zj,l (t), (l − 1)! l=1

t (−τ )m−l Hj,m [ e−λj τ f (τ )dτ + δl,m u0 ], (m − l)! 0

δl,m Kronecker’s delta. In case z(t) is bounded we must necessarily have zj,l (∞) = 0 for all j, l, i.e. ∞ m=l

Hj,m [

1 fˆ(m−l) (λj ) + δl,m u0 ] = 0 (m − l)!

for all l, j.

If all singularities of H(λ) are simple poles these conditions reduce to Hj,1 (u0 + fˆ(λj )) = 0,

j = 1, . . . , N,

and in both cases they are necessary and sufficient. Proposition 11.8 In addition to the assumptions of Theorem 11.4, suppose Σ0 = {λ1 , . . . , λN }, and let Hj,m denote the coefficients in the Laurent expansion of H(λ) at λ = λj in B(X).


303

Then, given u0 ∈ X, f ∈ Cub (X), u(t) defined by (11.35) is bounded on ⺢+ if and only if the compatibility conditions ∞ m=0

Hj,m+l [

1 ˆ(m) f (λj ) + δ0,m u0 ] = 0 m!

for all l ∈ ⺞, j ∈ {1, . . . , N }

are satisfied, where δl,m = 1 for l = m, δl,m = 0 otherwise. The proof of the sufficiency part follows by a direct estimation of z(t), where zj,l (t) can be replaced by zj,l (t) − zj,l (∞), since the compatilibity condition is equivalent to zj,l (∞) = 0. The details are left to the reader. 11.7 Comments a) The concept of homogeneous spaces as defined in Section 11.1 is the direct extension of the definition given in Katznelson [194] to the vector-valued case. The spaces Λ(X) have been employed in Pr¨ uss [271], [275] before. b) Admissibility of function spaces is a familiar topic in the theory of Volterra equations in finite dimensions; see e.g. the monographs of Miller [242], Corduneanu [60], and Gripenberg, Londen and Staffans [156]. In infinite dimensions, this concept has been introduced for Volterra equations with main part A, a closed linear operator in X with dense domain, and Y = XA in Pr¨ uss [271]; see also Pr¨ uss [275]. c) The results in Section 11.2, 11.3, and the first part of Section 11.4 are extensions of Pr¨ uss [271], [275], as are Theorems 11.3 and 11.4; they have been published recently in Pr¨ uss [280]. For the results on weakly almost periodic solutions see Pr¨ uss and Ruess [282]. d) In Staffans [314], 2π-periodic L2 -solutions of da ∗ u˙ + db ∗ u + dc ∗ Au = f in case X is a Hilbert space are studied, where a, b, c are of bounded variation on ⺢. In our terminology, this paper contains the full characterization of admissability + of L2Λ (X), where Λ = ⺪, in terms of boundedness properties of H(iρ) = (iρda(ρ) −1 db(ρ) + dc(ρ)A) on ⺪, like Proposition 11.5. In Da Prato and Lunardi [71] periodic solutions of u˙ = Au + B ∗ u + f are studied, where A generates an analytic semigroup in X and B ∈ L1 (⺢+ ; B(XA , X)) subject to very strong regularity assumptions. e) For the concept of scalar-valued almost periodic functions see the classical monograph of Bohr [27]; the definition given in Section 11.4, however, is due to Bochner who also extended this notion to the vector-valued case. The decomposition of a.a.p. functions was first obtained by Fréchet [119]. As a general reference for AP (X), we refer to Amerio and Prouse [9] and to Levitan and Zhikov [214].

304


The notion of weakly almost periodic function was introduced by Eberlein [107]. It should be observed that the definition due to Eberlein is different from that given in Amerio and Prouse. However, it seems that Eberlein’s concept is more appropriate for evolutionary problems, as recent results about w.a.p. solutions and ergodicity of nonlinear evolution equations show; see Ruess and Summers [294], [295], [296], [297], [298] and the survey paper [293]. The paper of Rosenblatt, Ruess and Sentilles [292] contains very interesting results about properties of W0+ . f) Not much seems to be known about the structure of the solution set of the homogeneous equation (11.3) with f = 0. In the simplest case Λ0 = {ρ1 , . . . , ρn }, the relation σ(u) ⊂ Λ0 implies via Proposition 0.5 that u(t) is of the form u(t) =

n

ak (t) exp(iρk ),

t ∈ ⺢,

k=1

where the ak (t) are X-valued polynomials. If in addition u belongs to any homogeneous space then the aj are constant, i.e. u(t) is a trigonometric polynomial with j )). Much further study is necessary to extend the coefficients aj ∈ N (iρj (I − A(iρ results which are known in finite dimensions to the infinite dimensional case. The same remark applies to problems with resonance, i.e. Λ ∩ Λ0 = ∅. For the latter the only reference is Miller and Wheeler [245] where for equations of scalar type in a Hilbert space with negative definite operator results on existence of an ⺢-kernel G(t) are proved, when Λ ∩ Λ0 consists of finitely many points and certain moment conditions hold. g) A theory similar to that presented here can be developed for equations of norder, i.e. for n dAk ∗ u(n−k) + f, (11.43) u(k) − A0 ∗ u(n) = k=1 1

where A0 ∈ L (⺢+ ; B(Y, X)), and Ak ∈ BV (⺢+ ; B(Y, X)), k = 1, . . . , n, and n ∈ ⺞0 is arbitrary. We have restricted the exposition to n = 1 since this case is most frequent in applications. The main difference consists in the behaviour of H(iρ) at ρ = 0, as the formal representation of the Fourier transform of the solution of (11.43) shows. u (ρ) = H(iρ)(iρ)1−k f,

ρ ∈ ⺢.

(11.44)

n (0) = An (∞) ∈ B(Y, X) is invertible. Note that 0 ∈ Λ0 for (11.43) iff dA It is also possible to reduce a second order problem to a first order one, convolving the second order equation with e−t e0 (t) and integrating by parts. In Section 13 we shall apply this technique to the Timoshenko beam model. h) If (11.3) is of scalar type then the approach via sums of commuting linear operators is applicable, in particular the results of Da Prato and Grisvard [64],


305

and Dore and Venni [101], Pr¨ uss and Sohr [283] are available; cp. Section 8. This approach has been carried through in Clément and Da Prato [52] via Theorem 8.3 and its consequences for (11.43) with A0 (t) = c(t)A, where c ∈ L1 (⺢+ ) is completely positive and A generates an analytic semigroup. In Clément and Pr¨ uss [54] Theorem 8.4 has been applied to linear heat conduction with memory; see also the survey papers of Clément [44], [45]. Another application of the sum method for equations on the line is contained in Pr¨ uss [279]. In each of these papers it turns out that Λ0 = ∅.

12

Admissibility of Function Spaces

This section is concerned with sufficiency of the frequency domain conditions derived in Proposition 11.5 for admissibility of evolutionary integral equations on the line as well as on the existence of Λ-kernels and their properties. The main results cover subordinated equations, hyperbolic problems in Hilbert spaces, and parabolic problems in arbitrary spaces. The discussion includes also perturbation problems and maximal regularity on the line for parabolic problems. 12.1 Perturbations: Hyperbolic Case For Λ ⊂ ⺢ unbounded, admissibility of HΛ (X) in the hyperbolic case is in general quite difficult to prove. There are not many tools available for this at present, the Paley-Wiener lemma, subordination and Parseval’s theorem in the Hilbert space case are the almost only ones. The main difficulty is the lack of decay properties of H(iρ) as |ρ| → ∞ which is in contrast to the parabolic case, and for this reason also multiplier theory does not work well, except for H(X) = L2 (⺢; X), when X and Y are Hilbert spaces. In this section we consider again the problem u˙ = A0 ∗ u˙ + dA1 ∗ u + f,

(12.1)

on the line, where A0 ∈ L1 (⺢+ ; B(Y, X)),

A1 ∈ BV (⺢+ ; B(Y, X)),

(12.2)

and A(t) = A0 (t) +A1 (t), as in Section 11. The spectrum of (12.1) will be denoted by Λ0 , as before. We begin the discussion here with an application of the Paley-Wiener lemma to the perturbed equation u˙ = A0 ∗ u˙ + dA1 ∗ u + B0 ∗ u + b ∗ B1 ∗ u + f,

(12.3)

where the unperturbed equation (12.1) has some admissibility properties at infinity. Theorem 12.1 Suppose the kernels A0 , A1 satisfy (12.2), let B0 ∈ L1 (⺢+ ; B(X)), B1 ∈ L1 (⺢+ ; B(Y, X)), and b ∈ W(⺢). Let Λ ⊂ ⺢ be closed, H(X) homogeneous, and assume (i) there is R > 0 such that HΛR (X) is b-admissible for (12.1), where ΛR = {ρ ∈ Λ : |ρ| ≥ R}; (ii) the spectrum Λ1 of (12.3) satisfies Λ ∩ Λ1 = ∅. Then HΛ (X) = BMΛ0 (X) is b-admissible for (12.3) if H(X) = BM 0 (X). Moreover, if (12.1) admits a ΛR -kernel GR , integrable in B(X), b ∗ GR integrable in B(X, Y ), then there is a Λ-kernel GΛ for (12.3) such that GΛ −GR ∈ L1 (⺢; B(X)), b ∗ (GΛ − GR ) ∈ L1 (⺢; B(X, Y )). In this case HΛ (X) is b-admissible, for any homogeneous space H(X). J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_12, © Springer Basel 1993

306

12. Admissibility

307

Proof: Choose N ≥ R + 1 and let GN ∈ ∩m≥0 W m,1 (⺢; B(X, Y )) be a kernel for ¯N +2 for (12.3), according to Theorem 11.1; observe that this theorem ΛN = Λ ∩ B applies by assumption (ii). Choose a cut-off function ϕ˜ ∈ C0∞ (⺢), 0 ≤ ϕ˜ ≤ 1, such that ϕ˜ = 1 for |ρ| ≤ N + 1, ϕ˜ = 0 for |ρ| ≥ N + 2 and decompose any f ∈ HΛ (X) as f = ϕ ∗ f + (f − ϕ ∗ f ) = f1 + f2 . Then σ(f1 ) ⊂ ΛN and σ(f2 ) ⊂ Λ ∩ {ρ ∈ ⺢ : |ρ| ≥ N + 1} = Λ∞ , by Proposition 0.6. By virtue of Theorem 11.1, the admissibility of HΛ (X) is this way reduced to admissibility of HΛ∞ (X); we may therefore assume ΛN = ∅, where N ≥ R + 1 is sufficiently large, specified later. Let G0 denote the solution operator for (12.1) in HΛ (X) = BMΛ0 (X) which exists and belongs to BHΛ (X), with R0 = b ∗ G0 ∈ B(BMΛ0 (X), BMΛ0 (Y )), by assumption (i). Then (12.3) may be rewritten as the fixed point equation u = G0 f + G0 B0 ∗ u + G0 B1 ∗ b ∗ u

(12.4)

or equivalently with u = G0 v, v = f + B0 ∗ G0 v + B1 ∗ R0 v.

(12.5)

Thus the solution operators G and b ∗ G for (12.3) are given by G = G0 (I − B0 ∗ G0 − B1 ∗ R0 )−1 ,

R = R0 (I − B0 ∗ G0 − B1 ∗ R0 )−1 ,

and to prove b-admissibility of HΛ (X) it is therefore sufficient to show |B0 ∗ G0 + B1 ∗ R0 | < 1. For this purpose consider the Fejer approximations BjN (t)

1 = 2π

N −N

ˆj (iρ)eiρt dρ, (1 − |ρ|/N )B

t ∈ ⺢, j = 0, 1;

since B0 ∈ L1 (⺢; B(X)) we have B0N → B0 in L1 (⺢; B(X)), and similarly B1N → B1 in L1 (⺢; B(Y, X)). Now for f ∈ BMΛ0 (X) we have ˜0N ⊂ σ(f ) ∩ B ¯N (0) ⊂ ΛN = ∅, σ(B0N ∗ f ) ⊂ σ(f ) ∩ supp B hence B0N ∗ f = 0, and similarly B1N ∗ g = 0 for g ∈ BMΛ0 (Y ). This implies |B0 ∗ G0 + B1 ∗ R0 | ≤ |B0 − B0N |1 |G0 | + |B1 − B1N |1 |R0 | < 1, provided N is sufficiently large. Note that the norms of G0 and R0 do not depend on N ≥ R + 1. If there exists a ΛR -kernel GR which is integrable in B(X), b ∗ GR integrable in B(X, Y ), then K = B0 ∗ GR + B1 ∗ b ∗ GR belongs to L1 (⺢; B(X)). By the

308


Paley-Wiener lemma, Theorem 0.6, there is a resolvent kernel R ∈ L1 (⺢; B(X)) for K. It follows readily from (12.4) that then GΛ = GR + GR ∗ R is a Λ-kernel for (12.4) with the properties asserted in Theorem 12.1. 2

Unfortunately, there are two shortcomings of Theorem 12.1 which limit its applicability. The first one is only conceptual, and could be removed, however, the second is of a more principal nature. For the first defect, recall that b-admissibility of the perturbed equation is only asserted in case H(X) = BM 0 (X). The reason for this is the fact that we do not know whether B1 ∗ b ∗ G0 leaves invariant HΛ (X) whenever this space is badmissible for the unperturbed equation. This is due to our definition of a solution; if a strong solution in H(X) would also require u ∈ H(Y ) this defect would not occur. However, the problem is that there is no ‘natural’ map X → H(X), so when considering a homogeneous space H(X), the ‘corresponding’ space H(Y ) is not at all clear to exist. If one restricts attention to such homogeneous spaces as Lp (X) there would be no problem. Note that in case B1 = 0 the result is true for any H(X). The second shortcoming of Theorem 12.1 is the assumption of integrability of GR and b ∗ GR , rather than their strong integrability. The reason for this is that it is not clear whether B0 ∗ GR and B1 ∗ b ∗ GR are uniformly integrable which is needed for the Paley-Wiener lemma. This can be easily shown in case GR and b ∗ GR are integrable rather than strongly integrable. 12.2 Subordinated Equations Next consider the equation of scalar type u˙ = a0 ∗ Au˙ + da1 ∗ Au + f,

(12.6)

where a0 ∈ L1 (⺢+ ), a1 ∈ BV (⺢+ ) are subordinate to a completely positive function c(t) according to a(t) = a0 (t) + a1 (t), and a ˆ(λ) = ˆb(1/ˆ c(λ)),

λ > 0,

(12.7)

∞

where 0 |b(t)|e−βt dt < ∞ for some β ∈ ⺢. Here the creep function k(t) associated with c(t) is given by k(t) = κ + ωt + 0

t

k1 (τ )dτ,

t > 0.

(12.8)

Let A be a closed linear operator in X with dense domain and let Sb (t) denote the resolvent of t v(t) = g(t) + b(t − τ )Av(τ )dτ, t > 0, (12.9) 0

12. Admissibility

309

and suppose

ωb = limt→∞ t−1 log |Sb (t)| < ∞.

According to Theorem 4.1, there is a resolvent Sa (t) for the equation subordinate to (12.9), and the growth bound for Sa (t) is not greater than ωa defined by ωa = cˆ−1 (1/ωb ) if ωb > 1/ω,

ωa = 0 otherwise.

(12.10)

We set again α = ω + k1 (0+), and as in Section 10.3 assume k1 ∈ W 1,1 (0, ∞), to be able to apply Proposition 4.9 and the structure theorem for subordinated resolvents, Theorem 4.4. Observe that if α > ωb then the spectrum Λ0 of (12.6) is compact; in fact 1/ˆb(z) ∈ ρ(A) for Re z > ωb by Theorem 1.3’ and Re 1/ˆ c(iρ) → α as |ρ| → ∞, hence 1/ˆ a(iρ) ∈ ρ(A) for |ρ| ≥ R, say. Theorem 12.2 Assume either of the following. (i) 1/κ + α = ∞; the function g(z) = z(ˆb(η + z)/ˆb(z) − 1) is locally analytic at ∞, for some η > max{β, ωb }; (ii) 1/κ+α < ∞, α > ωb ; the function g(z, w) = (ˆb(z)/ˆb(z +zw)−1)/w is analytic at (∞, 0). If Λ ⊂ ⺢ is closed such that Λ ∩ Λ0 = ∅, then there is a Λ-kernel GΛ (t) for (12.6). GΛ is of the form GΛ (t) = Sb (t/κ)e−αt/κ e0 (t) + G0 (t),

t ∈ ⺢,

(12.11)

where G0 ∈ L1 (⺢; B(X)) ∩ L∞ (⺢; B(X)) is B(X)-continuous on ⺢ \ {0}, even on ⺢ in case (ii), lim|t|→∞ |GΛ (t)|B(X) = 0, and the jump relation lim GΛ (t)x − lim GΛ (t)x = x,

t→0+

t→0−

for all x ∈ X,

(12.12)

is satisfied. Moreover, if rμ ∈ L1 (⺢) is such that r˜μ (ρ) =

a ˆ(iρ) , 1 + μˆ a(iρ)

ρ ∈ ⺢,

(12.13)

then HΛ (X) is rμ -admissible, for any homogeneous space H(X). Proof: As in the proof of Theorem 12.1 we may assume Λ ∩ BR (0) = ∅ where R can be chosen arbitrarily large, thanks to Theorem 11.1. Since Λ0 ⊂ ⺢ is compact ¯R−5 (0), and then w.l.o.g. as mentioned before, choose R so large that Λ0 ⊂ B ∞ Λ = ⺢ \ BR (0). Choose a cut-off function χ0 ∈ C (⺢) such that χ0 = 1 for |ρ| ≥ R − 1, χ0 = 0 for |ρ| ≤ R − 2, 0 ≤ χ ≤ 1 elsewhere; then χ0 = 1 − ψ˜0 , where ψ0 ∈ L1 (⺢). Choose another cut-off function χ1 ∈ C ∞ (⺢) such that χ1 = 1 on |ρ| ≥ R − 3, χ1 = 0 for |ρ| ≤ R − 4, 0 ≤ χ1 ≤ 1 elsewhere; we have χ1 = 1 − ψ˜1 for some ψ1 ∈ L1 (⺢). GΛ (t) will be constructed as in the proofs of Theorems 10.3 and 10.4 via the Paley-Wiener lemma applied to the convolution equations χ0 H = χ0 ϕη Hη − (χ1 ψη Hη )χ0 H,

(12.14)

310


corresponding to (10.21) in case (i), respectively, χ0 H0 = χ0 (ϕ1 + ϕ0 H1 )H1 + (χ1 ϕ0 H1 )χ0 H0 ,

(12.15)

which is the equivalent of (10.23) here in case (ii). The same notation employed there is deliberately also used here. The Paley-Wiener conditions are satisfied since |Hη (iρ)|B(X) → 0 in case (i), and ϕ0 (iρ) = ˆb0 (iρ) → 0 in case (ii), as |ρ| → ∞, provided R has been chosen large enough. Then in case (i), GΛ is given ˜ Λ = χ0 H, and G ˜ 0 = χ0 H0 +ϕ0 H1 in case (ii). The remaining properties of GΛ by G follow easily from the properties of Hη resp. H1 . Finally, to prove rμ -admissibility of HΛ (X), observe that GΛ is integrable in B(X), and ˜Λ A˜ rμ G

= (1 + μˆ a)−1 χ0 a ÂH = (1 − μ˜ rμ )χ0 (H − 1/iρ) ˜ Λ − (1 − μ˜ ˜μ, = (1 − μ˜ rμ )G rμ )χ0 /iρ = R

for some integrable function Rμ (t). 2

Observe that GΛ (t) satisfies GΛ x = A(a ∗ GΛ x) + e0 x − 1 ∗ ϕ0 x,

x ∈ X,

(12.16)

by uniqueness of the Fourier transform and closedness of A. GΛ (t) evidently commutes with A, and (12.16) implies d (GΛ x − h0 x − Aa0 ∗ GΛ x) = da1 ∗ AGΛ x − ϕ0 x, dt

x ∈ D(A).

(12.17)

Combining Theorem 12.2 with the perturbation result Theorem 12.1 now leads to sufficient conditions for admissibility of equations with main part. We leave the formulation of such a result to the reader. 12.3 Admissibility in Hilbert Spaces If X and Y are both Hilbert spaces one may use Parseval’s theorem to obtain admissibility of homogeneous spaces. The simplest result in this direction is certainly the following which concerns admissibility of L2Λ (X). Proposition 12.1 Suppose X, Y are Hilbert spaces, Λ ⊂ ⺢ is closed, and b ∈ W(⺢). Then L2Λ (X) is b-admissible if Λ ∩ Λ0 = ∅ and |H(iρ)|B(X) + |˜b(ρ)||H(iρ)|B(X,Y ) ≤ M, for some constant M ≥ 1. This is an easy consequence of the representation u ˜(ρ) = H(iρ)f˜(ρ),

ρ ∈ Λ,

ρ ∈ Λ,

(12.18)

12. Admissibility

311

of the solution of (12.1) for f ∈ L2Λ (X) with σ(f ) compact, and Parseval’s theorem; cp. Proposition 11.6. The details are left to the reader. If one imposes mild regularity restrictions on A(t) and the function b(t), and assumes existence of an exponentially bounded b-regular resolvent S(t), the full converse of Proposition 11.5 can be obtained. The resulting theorem contains as a special case Gearhart’s characterization of the spectrum of C0 -semigroups in Hilbert spaces. Theorem 12.3 Suppose X, Y are Hilbert spaces, Λ ⊂ ⺢ is closed, and let b ∈ L1 (⺢+ ) be 1-regular and such that ˆb(0) = 0. Assume that (6.1) admits a b-regular pseudo-resolvent, exponentially bounded, and in addition |λAˆ (λ)|B(Y,X) ≤ C|ˆb(λ)|,

Re λ > 0, |λ| ≥ R,

(12.19)

for some constants R, C > 0. Then Λ(X) is b-admissible if and only if Λ ∩ Λ0 = ∅ and (12.18) holds. In this case, for each x ∈ X there is hx ∈ L1 (X) ∩ L2 (X) such x (ρ) = H(iρ)x for all ρ ∈ Λ. Moreover, if Λ0 is that b ∗ hx ∈ L1 (Y ) ∩ L2 (Y ) and h compact, then there is a Λ-kernel GΛ : ⺢ → B(X), strongly continuous for t = 0, such that GΛ (·)x ∈ L1 (⺢) ∩ L∞ (⺢), GΛ (0+)x − GΛ (0−)x = x, lim|t|→∞ GΛ (t)x = ˜ Λ (ρ)x = H(iρ)x on Λ, for each x ∈ X. 0, and G Proof: The necessity part follows from Proposition 11.5. To prove the sufficiency ¯N (0) = ∅, where of (12.18) we may assume, as in the proof of Theorem 12.1, Λ ∩ B N > 0 can be arbitrarily large. ¯ε (0) with M replaced by (a) First we show that (12.18) holds also on Λε = Λ + B 4M , provided ε > 0 is chosen small enough. For this purpose consider the identity ˆ I − A(iρ)

ˆ 0 ) − A(iρ))H(iρ ˆ ˆ = [I + iρ0 (A(iρ 0 )](I − A(iρ0 )) ˆ = [I + K(ρ, ρ0 )](I − A(iρ0 )), ρ0 ∈ Λ, |ρ − ρ0 | ≤ ε;

we show |K(ρ, ρ0 )| ≤ 1/2 whenever ρ0 ∈ Λ, |ρ − ρ0 | ≤ ε, provided ε > 0 is sufficiently small. This then implies Λ0 ∩ Λε = ∅ and (12.18) with M replaced by 4M . From 1-regularity of b(t) we obtain the estimate ρ ρ0 +ε c |ˆb(iρ)| ≤ |ˆb(iρ0 )| + |ˆb (iτ )|dτ ≤ |ˆb(iρ0 )| + |ˆb(iτ )|dτ, |ρ0 | − ε ρ0 −ε ρ0 hence

ρ0 +ε ρ0 −ε

|ˆb(is)|ds ≤ 2ε(1 −

2cε −1 ˆ ) |b(iρ0 )| ≤ 4ε|ˆb(iρ0 )|, |ρ0 | − ε

if ρ0 ∈ Λ and ε ≤ N/(1 + 4c). This yields with (12.19) ρ ˆ |K(ρ, ρ0 )|B(X) ≤ |ρ0 | |dA(is)| B(Y,X) · |H(iρ0 )|B(X,Y ) ρ0

≤

CM |ρ0 | |ρ0 | − ε

ρ0 +ε ρ0 −ε

|ˆb(is)|ds ≤ 8CM ε,

312


for each ρ0 ∈ Λ, |ρ − ρ0 | ≤ ε, provided ε > 0 is small enough; recall |ρ0 | ≥ N . (b) By assumption, (6.1) admits a resolvent S(t) such that |S(t)|B(X) + |(b ∗ S)(t)|B(X,Y ) ≤ M0 e(ω−1)t ,

t ≥ 0,

for some constants M0 , ω > 0. But then the function u(t) = e−ωt S(t)xe0 (t) belongs to L2 (X) for each x ∈ X, and similarly v(t) = e−ωt (b∗S)(t)xe0 (t) belongs to L2 (Y ) ˆ + iρ)x = H(ω + iρ)x for each x ∈ X. By Parseval’s theorem therefore u ˜(ρ) = S(ω 2 ˆ belongs to L (X) as well, and similarly v˜(ρ) = b(ω + iρ)H(ω + iρ)x is in L2 (Y ) for every x ∈ X. (c) Choose a function ϕ ∈ C ∞ (⺢) such that 0 ≤ ϕ ≤ 1, |ϕ |∞ < ∞, ϕ ≡ 1 on Λε/3 , ϕ ≡ 0 on ⺢\Λ2ε/3 , and consider the identity H(iρ)x = (1 +

ω ˆ ˆ + iρ)]H(ω + iρ)x, )H(ω + iρ)x + H(iρ)(ω + iρ)[A(iρ) − A(ω iρ

which after multiplication with ϕ(ρ) yields ω (ω + iρ) ˆ ˆ + iρ)]˜ )ϕ(ρ)˜ u(ρ) + ϕ(ρ)H(iρ) [A(iρ) − A(ω v (ρ). ˆb(ω + iρ) iρ (12.20) Since u ˜ ∈ L2 (X), v˜ ∈ L2 (Y ) and 0 ∈ supp ϕ there follows ϕ(ρ)H(iρ)x ∈ L2 (X) and ˆ similarly ϕ(ρ)ˆb(iρ)H(iρ)x ∈ L2 (Y ), provided ˆb(iρ)/ˆb(ω + iρ) and (ω + iρ)[A(iρ) − ˆ ˆ A(ω + iρ)]/b(ω + iρ) are bounded for ρ ∈ Λε . To prove this we employ again 1-regularity of b(t) and (12.19). The identity ω ˆb(iρ + s) = ˆb(iρ + ω) − ˆb (iρ + τ )dτ ϕ(ρ)H(iρ)x = (1 +

s

yields with 1-regularity c |ˆb(iρ + s)| ≤ |ˆb(iρ + ω)| + |ρ| hence

ω 0

|ˆb(iρ + τ )|dτ ≤

ω 0

|ˆb(iρ + τ )|dτ,

Nω ˆ |b(iρ + ω)|, N − cω

0 ≤ s ≤ ω,

|ρ| > N,

(12.21)

provided N > cω; in particular ˆb(iρ)/ˆb(ω + iρ) is bounded on Λε . On the other hand, (12.19) and (12.21) yield ω ω C

ˆ ˆ ˆ |A(iρ) − A(ω + iρ)|B(Y,X) ≤ |A (s + iρ)|ds ≤ |ˆb(s + iρ)|ds |ρ| 0 0 C Nω ˆ ≤ · |b(ω + iρ)|, ρ ∈ Λε , |ρ| N − cω ˆ ˆ + iρ)]/ˆb(ω + iρ) is bounded on Λε as well. Thus, by hence (ω + iρ)[A(iρ) − A(ω Parseval’s theorem, for each x ∈ X there is hx ∈ L2 (X) such that b ∗ hx ∈ L2 (Y ) x (ρ) = H(iρ)x for ρ ∈ Λε . x (ρ) = ϕ(ρ)H(iρ)x, in particular h and h

12. Admissibility

313

(d) Next we show hx ∈ L1 (X) and b ∗ hx ∈ L1 (Y ) for each x ∈ X. For this ˆ observe first that in virtue of (12.19), A(iρ) is not only Lipschitz on Λε but also strongly differentiable a.e., since X and Y are Hilbert spaces, which enjoy the x therefore yields Radon-Nikodym property. Differentation of h

x (ρ) = h =

ϕ (ρ)H(iρ)x + iϕ(ρ)H (iρ)x ϕ (ρ)H(iρ)x − ϕ(ρ)H(iρ)x/ρ + iϕ(ρ)H(iρ)iρAˆ (iρ)H(iρ)x;

since 0 ∈ supp ϕ, ϕ is bounded and supp ϕ ⊂ Λε , by (12.19) we see that

x ∈ L2 (X), hence thx ∈ L2 (X) by Parseval’s theorem. Similarly, employing h 1-regularity of b(t) once more we obtain also t · b ∗ hx ∈ L2 (Y ). But then 1 √ |hx |1 = |hx (t)|dt + |hx (t)t|dt/|t| ≤ 2(|hx |2 + |thx |2 ) < ∞, −1

|t|≥1

i.e. hx ∈ L1 (X); similarly b ∗ hx ∈ L1 (Y ) for each x ∈ X. (e) Next we apply a duality argument to obtain h∗x ∈ L1 (X) ∩ L2 (X) such that ∗ (−ρ) = ϕ(ρ)H(iρ)∗ x for each x ∈ X. For this purpose observe that S ∗ (t)x h x is weakly continuous and strongly measurable on ⺢+ ; the latter follows again from the Radon-Nikodym property. Therefore H(ω + iρ)∗ x belongs to L2 (X) for ˆ ˆ + iρ))/ˆb(iρ), as above each x ∈ X, and by boundedness of (ω + iρ)(A(iρ) − A(ω we obtain also ϕ(ρ)H(iρ)∗ x ∈ L2 (X), for each x ∈ X. Also by duality we get (ϕ(ρ)H(iρ)∗ x) ∈ L2 (X), hence as in (c) and (d) there is h∗x ∈ L1 (X) ∩ L2 (X) ∗ (−ρ) = ϕ(ρ)H(iρ)∗ x, ρ ∈ ⺢, x ∈ X. such that h x By similar arguments, for each y ∈ Y there is gy∗ ∈ L1 (X) ∩ L2 (X) such that gy∗ (−ρ) = ϕ(ρ)(ˆb(iρ)H(iρ))∗ y. (f) The mappings x → hx , x → h∗x from X to L1 (X) are linear and closed; hence by the closed graph theorem, there is a constant C > 0 such that |hx |1 + |h∗x |1 ≤ C|x|X ,

x ∈ X.

Similarly, |gy∗ |1 ≤ C|y|Y , y ∈ Y, and |b ∗ hx |L1 (Y ) ≤ C|x|X , x ∈ X. To prove boundedness of the solution operator G from Λ(X) to Cub (X) and of b ∗ G from Λ(X) to Cub (Y ) it is enough to verify the relations ∞ (x, Gf (t))X = (h∗x (t − τ ), f (τ ))X dτ, t ∈ ⺢, (12.22) −∞

and (y, b ∗ Gf (t))Y =

∞

−∞

(gy∗ (t − τ ), f (τ ))X dτ,

t ∈ ⺢,

(12.23)

for each f ∈ Λ(X) such that σ(f ) is compact. In fact, (12.22) and (12.23) yield the estimates |Gf |L∞ (X) | ≤ C|f |L∞ (X) and |b ∗ Gf |L∞ (Y ) | ≤ C|f |L∞ (X) ,

314


for each f ∈ Λ(X) with compact spectrum, which by Proposition 11.5 is sufficient for b-admissibility of Λ(X). So let such f be given, choose another cutoff function ϕ˜0 ∈ C0∞ (⺢), which is real and symmetric, such that ϕ˜0 ≡ 1 on a neighborhood of σ(f ), and let ˜ 0 (ρ) = G0 ∈ W ∞,1 (⺢; B(X, Y )) denote the kernel from Lemma 11.1 such that G ∗ (ρ)x = G ˜ 0 (−ρ)∗ x = ϕ(ρ)H(iρ) ˜ ϕ˜0 (ρ), ρ ∈ ⺢. Then with ϕ0 ∗ f = f and G 0 ∗ (ρ), i.e. G∗ x = ϕ0 ∗ h∗ , we obtain ϕ˜0 (ρ)h x 0 x (x, Gf (t))X

∞ = (x, (G0 ∗ f )(t))X = (G∗0 (t − τ )x, f(τ ))X dτ −∞ ∞ ∞ ∗ = (hx (t − τ ), (ϕ0 ∗ f )(τ ))X dτ = (h∗x (t − τ ), f (τ ))X dτ, −∞

−∞

i.e. (12.22) holds. Similarly, one also obtains (12.23). (g) Finally, suppose Λ0 is compact; w.o.l.g. we then may choose ϕ(ρ) ≡ 1 for |ρ| sufficiently large, say |ρ| ≥ N 0 . Then 1 − ϕ ∈ C0∞ (⺢), ϕ(0) = 0, hence 1 − ϕ = ψ˜ ∞ for some ψ ∈ W ∞,1 (⺢) with −∞ ψ(t)dt = 1. Then the decomposition of the first term in (12.20), ϕ(ρ)(1 +

ω ϕ(ρ) ˜ )H(ω + iρ)x = H(ω + iρ)x − ψ(ρ)H(ω + iρ)x + ω H(ω + iρ)x, iρ iρ

shows that this term is the Fourier transform of a bounded function w1 (t) which is continuous for t = 0, satisfies the jump relation limt→0+ w1 (t) − limt→0− w1 (t) = w1 (0+) − w1 (0−) = x as well as lim|t|→∞ w1 (t) = 0. On the other hand, the second term in (12.20) can be written in the form H(ω + iρ)

(ω + iρ) ˆ ˆ + iρ)]ϕ(ρ)ˆb(iρ)H(iρ)x = H(ω + iρ)w˜2 (ρ), [A(iρ) − A(ω ˆb(iρ)

where w2 ∈ L2 (X), hence is the Fourier transform of Se−ω· e0 ∗ w2 , which belongs to C0 (⺢; X). The proof is complete. 2

From the proof of Theorem 12.3 it follows that in case A(t) decomposes as A(t) = B(t) + C(t), where B ∈ L1loc (⺢+ ; B(Y, X)) and C ∈ L1loc (⺢+ ; B(X)), only B(t) need be subject to (12.19). For C(t) the weaker condition |λCˆ (λ)| ≤ M,

Re λ > 0, |λ| ≥ R,

is sufficient. Observe that (12.24) is implied by

∞ 0

t|dC(t)|B(X) < ∞,

(12.24)

12. Admissibility

315

whenever C ∈ BVloc ((0, ∞); B(X)). This remark will turn out to be quite useful in Section 13. By an interpolation argument the admissibility assertion in Theorem 12.3 can be extended to LpΛ (X), for all 1 ≤ p < ∞. Corollary 12.1 Let the assumptions of Theorem 12.3 be satisfied and let (12.18) be fullfilled. Then LpΛ (X) is b-admissible, 1 ≤ p < ∞. n Proof: For a simple function f = i=1 xi χi , where xi ∈ X and χi are characteristic functions of disjoint measurable sets of finite measure, we define Gf =

n

h x i ∗ χi .

i=1

We then have obviously the estimate |Gf |1 ≤ C|f |1 ; cp. the first step in (f) of the proof of Theorem 12.4. On the other hand, as there we also have |Gf |∞ ≤ C|f |∞ , hence by the vector-valued Riesz-Thorin interpolation theorem (see e.g. Bergh and L¨ ofstr¨ om [22]) G ∈ B(Lp (X)) for each 1 ≤ p < ∞. Similarly, b ∗ G ∈ p B(L (X), Lp (Y )), and therefore LpΛ (X) is b-admissible for every p ∈ [1, ∞). 2

Since for hyperbolic problems Λ0 need not be compact in general, it remains open whether in the situation of Theorem 12.3 a Λ-kernel GΛ always exists. Even if so, it seems to be very difficult to obtain pointwise estimates of GΛ (t), in particular to prove the integrability of GΛ . For this reason it also seems to be quite unclear whether Corollary 12.1 can be extended to homogeneous spaces other than LpΛ (X), e.g. it is not known whether BMΛ0 (X) is b-admissible, in this situation. Let us conclude this section with an illustrative application of Theorem 12.3 to obtain the characterization of the spectrum of a C0 -semigroup in a Hilbert space, i.e. Gearharts theorem mentioned above; see Section 12.7 for further comments. Example 12.1 Let A generate a C0 -semigroup eAt in the Hilbert space X. Choose Y = XA , b(t) = e−t e0 (t), Λ = (2π/τ )⺪. With A(t) ≡ A the assumptions of Theorem 12.3 are easily seen to be true; the resolvent S(t) is of course the C0 semigroup eAt . A moment of reflection shows also that Λ(X) is b-admissible if and only if the evolution equation u˙ = Au + f (12.25) admits a mild τ -periodic solution u, for each f ∈ Cub (X) τ -periodic. Via the variation of parameters formula the latter is equivalent to 1 ∈ ρ(eAτ ). Theorem 12.3 characterizes this property by {2πni/τ : n ∈ ⺪} ⊂ ρ(A), sup{|(2πni/τ − A)−1 | : n ∈ ⺪} < ∞,

(12.26)

which is precisely Gearhart’s condition. 12.4 Λ-kernels for Parabolic Problems Consider again (12.1) assuming the decomposition (12.2). In Section 11.3 we have

316


seen that for compact Λ ⊂ ⺢, the spectral condition Λ ∩ Λ0 = ∅ is sufficient for badmissibility of HΛ (X), for any homogeneous space H(X) and any b ∈ W(⺢). For problems of parabolic type this result can be extended to noncompact Λ, however, b will no longer be arbitrary then. We begin with the case when (7.1) admits an analytic resolvent. Theorem 12.4 Suppose (7.1) admits a weak analytic resolvent S(t) of type (ω0 , θ0 ), and let (AN4) with a = b ∈ W(⺢+ ) be satisfied; cp. Corollary 7.2. Let H(X) be homogeneous, and assume Λ ⊂ ⺢ is closed and such that Λ ∩ Λ0 = ∅. Then HΛ (X) is b-admissible, and there is a Λ-kernel GΛ ∈ L1 (⺢; B(X)), b ∗ GΛ ∈ L1 (⺢; B(X, Y )) for HΛ (X). Moreover, G0 (t) = GΛ (t) − S(t)e0 (t) admits extension to an entire function with values in B(X, Y ) of exponential growth, and |G0 (t)|B(X,Y ) ≤ Cε max(1, e(ω0 +ε)t ), t ∈ ⺢. There is ϕ0 ∈ C0∞ (⺢) such that G˙ 0 − ϕ0 = A0 ∗ G˙ 0 + dA1 ∗ G0 = G˙ 0 ∗ A0 + G0 ∗ dA1

on ⺢.

(12.27)

ˆ Proof: Corollaries 7.1 and 7.2 imply that the family H(λ) = S(λ) is holomorphic on Σ(ω0 , θ0 + π/2) with values in B(X, Y ) and for each ω > ω0 , θ < θ0 there is a constant C = C(ω, θ) such that |H(λ)|B(X) + |ˆb(λ)H(λ)|B(X,Y ) ≤ C/|λ − ω|,

λ ∈ Σ(ω, θ + π/2).

(12.28)

Therefore, Λ0 is compact, Λ0 ⊂ [−N0 , N0 ], where N0 = ω cot θ0 , and so d(Λ0 , Λ) > 0. As in the proof of Theorem 12.1 we may assume that Λ ∩ BN0 +3 (0) = ∅. Choose a cut-off function ψ(iρ) of class C ∞ such that ψ(iρ) = 1 for |ρ| ≥ N0 + 2, ψ(iρ) = 0 for |ρ| ≤ N0 + 1, and 0 ≤ ψ ≤ 1. Then ψ(iρ) − 1 is of class C0∞ (⺢), hence ψ(iρ) − 1 = ϕ˜0 (ρ), for some ϕ0 ∈ C ∞ (⺢) ∩ L1 (⺢). Set ψ(λ) = 1 for Re λ = 0, and choose a smooth curve Γ+ ⊂ Σ(ω0 , θ0 + π/2) ∩ {λ ∈ ⺓ : Re λ ≤ ω} running from i(N0 + 3) to −i(N0 + 3), ω > ω0 , let Γ0 = Γ+ ∪ [−i(N0 + 3), i(N0 + 3)], and define 1 G0 (z) = ψ(λ)H(λ)eλz dλ, z ∈ ⺓. (12.29) 2πi Γ0 Then G0 : ⺓ → B(X, Y ) is entire, |G0 (z)|B(X,Y ) ≤ Cω eω|z| for some Cω > 0, and |G(n) (t)|B(X,Y ) ≤ Cn for t ≤ 0, n ∈ ⺞0 . A direct computation shows that G˙ 0 − A0 ∗ G˙ 0 − dA1 ∗ G0 = G˙ 0 − G˙ 0 ∗ A0 − G0 ∗ dA1 ∞ 1 1 λt = ψ(λ)e dλ = (ψ(iρ) − 1)eiρt = ϕ0 (t), t ∈ ⺢, 2πi Γ0 2π −∞

i.e. (12.27) holds. Define GΛ (t) according to GΛ (t) = G0 (t) + S(t)e0 (t),

t ∈ ⺢;

(12.30)

12. Admissibility

317

then by boundedness of S(t), GΛ (t) is bounded in B(X), uniformly on each interval of the form (−∞, t0 ]. Cauchy’s theorem and (12.28) show that we have the representation N 1 GΛ (t) = lim ψ(iρ)H(iρ)eiρt dρ, t = 0. (12.31) 2π N →∞ −N To prove integrability of GΛ we integrate twice by parts. Since ψ vanishes on [−N0 − 1, N0 + 1], the function Φ(iρ) = ψ(iρ)H(iρ) is of class C ∞ on ⺢, and by Cauchy’s theorem we have |Φ(j) (ρ)| ≤ Mj /(1 + |ρ|)j+1 ,

for all ρ ∈ ⺢, j ∈ ⺞0 .

Integrating twice by parts we obtain ∞ 1 GΛ (t) = − Φ

(ρ)eiρt dρ, 2πt2 −∞

(12.32)

t ∈ ⺢,

hence GΛ ∈ L1 (⺢; B(X)). Replacing H(λ) by ˆb(λ)H(λ) in these arguments, we also obtain b ∗ GΛ ∈ L1 (⺢; B(X, Y )). Finally, we show that the kernel GΛ represents the solution operator G by (11.14); b-admissibility of any HΛ (X) is then an easy consequence of the integrability properties of GΛ and of Proposition 11.6. From the first resolvent equation S = I + A ∗ S and b-regularity of S(t) there follows by (12.27) an identity for R = b ∗ GΛ in B(X), R˙ = A0 ∗ R˙ + dA1 ∗ R + b ∗ ϕ0 + b. Hence u = R ∗ f is a solution of (12.1) with inhomogeneity b ∗ f + ϕ0 ∗ b ∗ f , for each f ∈ HΛ (X). However, since ϕ˜0 (ρ) = ψ(iρ) − 1 = 0 for ρ ∈ σ(f ), we have σ(ϕ0 ∗ f ) ⊂ σ(ϕ0 ) ∩ σ(f ) = ∅, hence ϕ0 ∗ f = 0. Thus u = R ∗ f = GΛ ∗ b ∗ f is a strong solution of (12.1) with inhomogeneity b ∗ f , and so (11.14) is valid. 2

The main property of H(iρ) which was used in the proof of Theorem 12.4 is the estimate (12.32). This decay property is also present in the case of weakly parabolic equations with weakly 2-regular kernels satisfying (NP4); see Section 7.2. Somewhat more generally we have Corollary 12.2 Suppose b ∈ L1 (⺢+ ) is 2-regular, b(0) = 0, and let the following conditions hold. ∈ B(Y, X) is bijective and H(λ) = (i) For each Re λ ≥ 0, |λ| ≥ R, I − A −1 (I − A(λ)) /λ satisfies for some R > 0 M |H(λ)|B(X) + |b(λ)H(λ)|B(X,Y ) ≤ , |λ|

for all Re λ ≥ 0, |λ| ≥ R.

(ii) There is a constant C > 0 such that (λ)|B(Y,X) + |λ2 A

(λ)|B(Y,X) ≤ |b(λ)|, |λA

for all Re λ ≥ 0, |λ| ≥ R.

318


Let H(X) be homogeneous, and assume Λ ⊂ ⺢ is closed and such that Λ ∩ Λ0 = ∅. Then the assertions of Theorem 12.4 are valid. For the proof observe only that H(λ) satisfies by (i) and (ii) the crucial estimate (12.32) and that after an exponential shift of size ω ≥ R the assumptions of Corollary 7.4 are satisfied, hence there is a b-regular weak resolvent S(t). Therefore the proof of Theorem 12.4 carries over. 12.5 Maximal Regularity on the Line As one might expect, for parabolic equations there are also maximal regularity results on the line. We again employ the notation introduced in Section 7.4, with J = ⺢; for α ∈ (0, 1), p ∈ [1, ∞), q ∈ [1, ∞] we let Bpα,q (X) = (Lp (X), W 1,p (X))α,q , denote the real interpolation space between Lp (X) and W 1,p (X) of order α and power q, and similarly for p = ∞ α,q 1 B∞ (X) = (Cub (X), Cub (X))α,q ;

see Section 7.4 for the definition of the norms. These spaces are easily seen to be homogeneous. Theorem 12.5 Let the assumptions of Theorem 12.4 or Corollary 12.2 be fulfilled. α,q Then for α ∈ (0, 1), p, q ∈ [1, ∞], the spaces Bp,Λ (X) are spaces of maximal α,q regularity for (12.1), i.e. if f ∈ Bp (X), σ(f ) ⊂ Λ, then the solution u = Gf = GΛ ∗ f of (12.1) satisfies u, u˙ ∈ Bpα,q (X) and b ∗ u, b ∗ u˙ ∈ Bpα,q (Y ). Proof: (a) Suppose the assumptions of Theorem 12.4 or Corollary 12.2 are fulfilled, and let GΛ (t) denote the corresponding Λ-kernel for the closed set Λ ⊂ ⺢; w.l.o.g. we may assume Λ = {ρ ∈ ⺢ : |ρ| ≥ R}, for R > 0 sufficiently large. Also, with some fixed ω > R + 1 we let Sω (t) = e−ωt e0 (t)S(t), t ∈ ⺢, where S(t) denotes the weak resolvent, which exists as was observed before. Then the results of Section 7.1 and 7.2 show that Sω satisfies |Sω (t)|B(X) + |tS˙ ω (t)|B(X) ≤ N e−t, t > 0, t−s t |S˙ ω (t) − S˙ ω (s)|B(X) ≤ N (1 + log ), t > s > 0; ts t−s

(12.33)

cp. (7.29). With bω (t) = e−ωt b(t), Tω = bω ∗ Sω is also subject to (12.33), where B(X) can be replaced by B(X, Y ). (b) Define G1 (t) by G1 (t) = GΛ (t) − Sω (t), t ∈ ⺢; (12.34) then (12.33) and uniform integrability of GΛ (t) yield G1 ∈ L1 (B(X)). We show 1 (ρ) and decompose as that G1 ∈ W 1,1 (B(X)). For this purpose consider iρG

12. Admissibility

319

follows 1 (ρ) = iρG

ψ(iρ)iρH(iρ) − iρH(ω + iρ)

=

ψ(iρ)(iρH(iρ) − (ω + iρ)H(ω + iρ)) + ωψ(iρ)H(ω + iρ)

+

(ψ(iρ) − 1)iρH(ω + iρ) ˜˙ 0 (ρ)S˜ω (ρ), Φ2 (ρ) + ω(1 + ϕ˜0 (ρ))S˜ω (ρ) + ϕ

=

(12.35)

where ψ and ϕ0 are as in the proof of Theorem 12.4. If the assumptions of Corollary 12.2 are satisfied we may apply the arguments following (12.21) in the proof of Theorem 12.3 to the result that −1 ˆ ˆ ˆ + iρ))(I − A(ω ˆ + iρ))−1 Φ2 (ρ) = ψ(iρ)(I − A(iρ)) (A(iρ) − A(ω

is twice differentiable and subject to |Φ2 (ρ)| + (1 + |ρ|)(|Φ 2 (ρ)| + |Φ

2 (ρ)|) ≤ M0 (1 + |ρ|)−1 ,

ρ ∈ ⺢.

(12.36)

In the situation of Theorem 12.4, Cauchy type estimates lead to (12.36) as well. Integrating by parts, (12.35) gives N 1 Φ2 (ρ)eiρt dρ N →∞ 2π −N ∞ ∞ −1 −1 = Φ 2 (ρ)(eiρt − 1)dρ = Φ

(ρ)eiρt dρ, 2πit −∞ 2πt2 −∞ 2 ∞ where the last integrals are absolutely convergent; observe that −∞ Φ 2 (ρ)dρ = 0 by (12.36). Thus G2 (t) is well-defined for t = 0, and direct estimates yield G2 (t) :=

lim

|G2 (t)| ≤ M1 min{1/t2 , 1 + log(1 + 1/|t|)},

t = 0,

hence G2 ∈ L1 (B(X)). But since the last two terms of (12.35) are the Fourier transforms of ω(Sω + ϕ0 ∗ Sω ) and ϕ˙ 0 ∗ Sω , respectively, we have obtained G˙ 1 ∈ L1 (B(X)) by uniqueness of the Fourier transform. Similarly one also proves b∗G1 ∈ L1 (B(X, Y )). (c) Now let f ∈ Bpα,q (X), σ(f ) ⊂ Λ, where α ∈ (0, 1), p, q ∈ [1, ∞], and let u = Gf = GΛ ∗ f = G1 ∗ f + Sω ∗ f denote the solution of (12.1). It is then not difficult to show by a limiting argument that u is differentiable a.e. and ∞ ∞ u(t) ˙ = G˙ 1 (τ )f (t − τ )dτ + S˙ ω (τ )(f (t − τ ) − f (t))dτ, for a.a. t ∈ ⺢, −∞

0

(12.37) holds. Since G˙ 1 ∈ L1 (B(X)) the first term leaves every homogeneous H(X) invariant, in particular the spaces Bpα,q (X). On the other hand, by means of (12.33) a minor modification of the proof of Theorem 7.5 shows that the last term has this property too. Therefore u, u˙ ∈ Bpα,q (X). Since Tω is subject to (12.33) with B(X) replaced by B(X, Y ) we obtain b ∗ u, b ∗ u˙ ∈ Bpα,q (Y ). The proof is complete. 2

320


12.6 Perturbations: Parabolic Case Via the usual fixed point argument, Theorem 12.5 can be extended to the perturbed problem u˙ − A0 ∗ u˙ − dA1 ∗ u = B0 ∗ u˙ + dB1 ∗ u + b ∗ (C0 ∗ u˙ + dC1 ∗ u) + f

(12.38)

where B0 ∈ L1 (⺢+ ; B(X)), B1 ∈ BV (⺢+ ; B(X)), C0 ∈ L1 (⺢+ ; B(Y, X)), and C1 ∈ BV (⺢+ ; B(Y, X)). Corollary 12.3 Let the assumptions of Theorem 12.4 or Corollary 12.2 be satisfied, where Λ0 is replaced by the spectrum Λ1 of the perturbed problem (12.38). α,q Then for each α ∈ (0, 1), p, q ∈ [1, ∞], the spaces Bp,Λ (X) are spaces of maximal regularity for (12.38). Proof: W.o.l.g. Λ = {ρ ∈ ⺢ : |ρ| ≥ R}, where R is sufficiently large. Let B0N (t) denote the Fejer approximation of B0 , i.e. N 1 |ρ| N B0 (t) = (1 − )B0 (ρ)eiρt dρ, t ∈ ⺢; 2π −N N B˙ 1N , C0N , C˙ 1N are defined similarly. Then by choosing N large enough |B0N − B0 |L1 (B(X)) + |e−t ∗ (dB1 − B˙ 1N )|L1 (B(X)) +|C0N − C0 |L1 (B(Y,X)) + |e−t ∗ (dC1 − C˙ 1N )|L1 (B(Y,X)) ≤ η(N ), N where η(N ) → 0 as N → ∞. Since supp B 0 ⊂ BN (0) we may replace B0 by N B0 − B0 in (12.38) and similarly C0 by C0 − C0N , in case σ(f ) ∩ BN +1 (0) = ∅. On the other hand, we may write dB1 ∗ u = (dB1 − B˙ 1N ) ∗ u = e−t ∗ (dB1 − B˙ 1N ) ∗ (u + u), ˙ and similarly the term dC1 ∗b∗u is treated. These considerations show that (12.38) can be transformed into the problem u˙ − A0 ∗ u˙ − dA1 ∗ u = B2 ∗ u˙ + B3 ∗ u + b ∗ (C2 ∗ u˙ + C3 ∗ u) + f,

(12.39)

where the L1 -norms of Bj and Cj are as small as we please, provided σ(f ) ∩ BN +1 (0) = ∅ for a sufficiently large N . This means that we have the fixed point problem u = Gf + G(B2 ∗ u˙ + B3 ∗ u) + G(C2 ∗ b ∗ u˙ + C3 ∗ b ∗ u), which by the contraction mapping principle in virtue of Theorem 12.5 is uniquely α,q solvable in each of the spaces Bp,Λ (X), provided R is large enough. 2

To construct a Λ-kernel GΛ for (12.38) consider the convolution equation GΛ = G0Λ +

d 0 G ∗ (B2 ∗ GΛ + C2 ∗ b ∗ GΛ ) + G0Λ ∗ (B3 ∗ GΛ + C3 ∗ b ∗ GΛ ); (12.40) dt Λ

12. Admissibility

321

since (12.39) is equivalent to (12.38) for Λ = {ρ ∈ ⺢ : |ρ| ≥ R}, R sufficiently large. Here G0Λ denotes the Λ-kernel for (12.1) according to Theorem 12.4 or Corollary 12.2. We are now looking for a solution GΛ ∈ L1 (B(X)) with b∗GΛ ∈ L1 (B(X, Y )). Note that the Paley-Wiener lemma Theorem 0.6 cannot be applied here, since there is no chance to prove that the involved kernel is L1 , without imposing extra assumptions at least. However, estimates (12.33) for Sω (t) and the estimate for G2 derived in (b) of the proof of Theorem 12.5 show α,∞ G0Λ ∈ Bp1/p,∞ (B(X)) ∩ B1 (B(X)); p∈(1,∞)

α 0.

(13.3)

Thus (13.1) is of hyperbolic type and almost synchronous, since 1 (λ) de e1 (0+) + e˙1 (λ) = = dk(λ), 1 (λ) a1 (0+) + a˙1 (λ) da

λ > 0,

for some k ∈ BVloc (⺢+ ) with k(0+) = limλ→∞ dk(λ) = e1 (0+)/a1 (0+) > 0. Let 2 2 X = L (0, 1) × L (0, 1) and Y = {(w, φ) ∈ W 2,2 ([0, 1])×W 2,2 ([0, 1]) : w(0) = φ(0) = φ (1) = w (1)+φ(1) = 0} J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8_13, © Springer Basel 1993

323

324


be equipped with their natural norms, and define A1 (t) ∈ B(Y, X) by means of a1 (t)∂x2 , a1 (t)∂x A1 (t) = , t ∈ ⺢+ . −γa1 (t)∂x , e1 (t)∂x2 − γa1 (t) Then with u = (w, φ)T , (13.1) can be rewritten as u ¨ = dA1 ∗ u + f,

t ∈ ⺢,

(13.4)

a second order problem. Apparently, (13.4) is not of the form (11.3), however, convolving (13.4) with c(t) = e−t e0 (t), e0 the Heaviside function, it becomes u˙ = c ∗ u˙ + c ∗ dA1 ∗ u + g,

t ∈ ⺢,

(13.5)

where g = c ∗ f , which is of the form (11.3). Since c ∈ W(⺢+ ) it is easily seen that (13.4) and (13.5) are equivalent. We want to apply Theorem 12.3 to obtain admissibility of Λ(X), as well as Corollary 12.1 for LpΛ (X), 1 ≤ p < ∞, where Λ ⊂ ⺢ closed is such that Λ ∩ Λ0 = ∅. For this purpose recall first from Section 9.1 that there is a b-regular exponentially bounded resolvent S(t) for the local versions of (13.4) and (13.5), where e.g. b(t) = tc(t) is a possible choice. Obviously b ∈ L1 (⺢+ ) is 1-regular and such that ˆb(0) = 0. To verify (12.19) for (13.5), observe that we have to show |λ[ˆ cAˆ1 ] (λ)|B(Y,X) ≤ C|ˆb(λ)|, and

|λˆ c (λ)| ≤ M,

Re λ > 0, |λ| ≥ R,

Re λ > 0, |λ| ≥ R,

by the remark following the proof of Theorem 12.3. The second of these inequalities is obvious, while the first follows from |λ2 a ˆ 1 (λ)| + |λ2 eˆ 1 (λ)| ≤ M0 , as an easy calculation shows; but the latter is a consequence of (13.3), since e.g. ∞ ∞ 2

|λ a ˆ1 (λ)| = |tda˙ 1 (λ) + 2a˙ 1 (λ)| ≤ tda˙ 1 (t) − 2 a˙ 1 (t)dt ≤ 3a1 (0+), 0

0

by convexity of a1 (t). Thus the assumptions of Theorem 12.3 are satisfied and it remains to compute Λ0 and to show boundedness of H(iρ) and ˆb(iρ)H(iρ) in B(X) resp. in B(X, Y ). 1 (λ) on This means that we have to study the invertibility properties of λ2 − dA τ the imaginary axis, i.e. for λ = iρ, ρ ∈ ⺢. So let u = (w, φ) , f = (g, ψ)τ and consider the equation (ρ2 + dA(iρ))u = f , i.e. ρ2 w

+

ρ2 φ + w(0) =

1 (iρ)w

+ da 1 (iρ)φ = g, da 1 (iρ)φ

− γ da 1 (iρ)(w + φ) = ψ, de

φ(0) = 0, φ (1) = w (1) + φ(1) = 0.

(13.6)

13. Applications and Complements

325

Taking the inner product in L2 (0, 1) of the first equation with w, integrating by parts and employing the boundary conditions we obtain the identity 1 (iρ)[|w |2 + (φ, w )] + (g, w); ρ2 |w|2 = da

(13.7)

similarly, the second equation yields after multiplication with φ 1 (iρ)|φ |2 + γ da 1 (iρ)[(w , φ) + |φ|2 ] + (ψ, φ). ρ2 |φ|2 = de

(13.8)

Here we used the notation (·, ·) for the inner product and | · | for the norm in L2 (0, 1). Multiplying (13.7) with γ and adding the result to (13.8) leads to 1 (iρ)|φ |2 + γ da 1 (iρ)|w + φ|2 + γ(g, w) + (ψ, φ), ρ2 (γ|w|2 + |φ|2 ) = de

(13.9)

and taking real and imaginary parts in this identity we finally obtain ρ2 (γ|w|2 + |φ|2 ) =

1 (iρ)|φ |2 + γ Re da 1 (iρ)|w + φ|2 Re de +γ Re (g, w) + Re (ψ, φ),

(13.10)

and 1 (iρ)|φ |2 + γ Im da 1 (iρ)|w + φ|2 + γ Im (g, w) + Im (ψ, φ). (13.11) 0 = Im de The properties (13.3) of e1 and a1 imply 1 (iρ) ≥ e∞ > 0 Re de 1 (iρ) ≥ 0 ρ Im de 1 (iρ) = e1 (0+) lim de

|ρ|→∞

1 (iρ) ≥ a∞ > 0. Re da 1 (iρ) ≥ 0, , ρ Im da 1 (iρ) = a1 (0+), , lim da ,

|ρ|→∞

1 (iρ) = 0 if and only if e1 (t) is as is easily verified. Moreover, for ρ > 0, Im de piecewise linear, with nodes only at t ∈ (2π/ρ)⺞. If e1 (t) = e∞ , then there is a 1 (iρ) = 0 if and only if ρ ∈ ρe ⺪. The smallest such ρ, say ρe > 0, and then Im de same is true of course for a1 , with a smallest ρa > 0, in case a1 (t) = a∞ . is invertible in B(X, Y ) or 0 is To compute Λ0 observe that either ρ2 + dA(iρ) 2 the latter is equivalent to ρ ∈ Λ0 . Let g = ψ = 0 an eigenvalue of ρ + dA(iρ); 1 (iρ) = 0; then (13.11) implies φ ≡ 0, hence φ ≡ 0 by (13.6), and assume Im de and then da1 (iρ)(w + φ) = 0, hence ρ2 w = 0, by (13.6) again. Therefore Im 1 (iρ) = 0, (13.11) 1 (iρ) = 0 implies ρ ∈ Λ0 . Similarly, if g = ψ = 0 and Im da de yields w + φ ≡ 0, and then by (13.6) again ρ2 w ≡ 0, i.e. w ≡ 0 and φ ≡ 0, i.e. ρ ∈ Λ0 . 1 (iρ) = Im da 1 (iρ) = 0, i.e. since 0 ∈ Λ, e1 (t) and Thus ρ ∈ Λ0 implies Im de a1 (t) must be of the special form explained above. In particular, if e1 or a1 is log-convex, or −e˙ 1 or −a˙ 1 is convex, then Λ0 = ∅.

326


1 (iρ) = 0, Im da 1 (iρ) = 0 for all ρ ∈ Λ Now let Λ ⊂ ⺢ be closed such that Im de and 1 (iρ) 1 (iρ) Re de Re da sup{ , : ρ ∈ Λ} = N < ∞. (13.12) 1 (iρ) ρ Im da 1 (iρ) ρ Im de Then (13.10) and (13.11) yield for |ρ| ≥ 1 ρ2 (γ|w|2 + |φ|2 ) ≤ γ Re (g, w) + Re (ψ, φ) − N ρ(γ Im (g, w) + Im (ψ, φ)) ≤ C1 |ρ|(γ|g| |w| + |ψ| |φ|), hence

ρ2 (γ|w|2 + |φ|2 ) ≤ C12 (γ|g|2 + |ψ|2 ),

ρ ∈ Λ, |ρ| ≥ 1.

(13.13)

1 (iρ) ≥ e∞ > 0, Re da 1 (iρ) ≥ a∞ > 0, With (13.10) this implies by Re de e∞ |φ |2 + γa∞ |w + φ|2 ≤ C22 (γ|g|2 + |ψ|2 ),

(13.14)

and (13.6) gives |w

|2 + |φ

|2 ≤ C32 (γ|g|2 + |ψ|2 )ρ2 ,

ρ ∈ Λ, |ρ| ≥ 1.

(13.15)

Estimates (13.13) and (13.15) yield 1 (iρ))−1 |B(X) + 1 |(ρ2 + dA 1 (iρ))−1 |B(X,Y ) ≤ C4 , |ρ(ρ2 + dA |ρ|

ρ ∈ Λ, |ρ| ≥ 1,

which implies |H(iρ)|B(X) + |ˆb(iρ)| |H(iρ)|B(X,Y ) ≤ C5 ,

ρ ∈ Λ,

(13.16)

for (13.5) . Invoking Theorem 12.3 then yields b-admissibility of Λ(X), and Corollary 12.1 that of LpΛ (X). We may now apply Theorem 11.2 or Corollary 11.2 and 11.3 to deduce results on periodic or almost periodic solutions of (13.1); the details of this, however, are left to the reader. Concluding we refer to the recent paper Desch and Grimmer [85] where a detailed analysis of the spectrum of the viscoelastic Timoshenko beam has been performed in case the relaxation moduli are completely monotonic. Under certain assumptions on the behaviour of their Laplace transforms for large frequencies it is shown that the spectrum of the viscoelastic Timoshenko beam asymptotically decomposes into the spectra of the two decoupled viscoelastic wave equations arising from (13.1) by neglecting the lower order terms da ∗ φx and γda ∗ (wx + φ). 13.2 Heat Conduction in Materials with Memory Consider the problem of heat conduction in materials with memory described in Section 5.3, where the rigid body Ω is assumed to be homogeneous, isotropic and bounded. The dynamic equations on the line are then given by dm ∗ θ˙ θ ∂θ ∂n

= dc ∗ Δθ + f,

t ∈ ⺢, x ∈ Ω,

= 0,

t ∈ ⺢, x ∈ Γb ,

= 0,

t ∈ ⺢, x ∈ Γf ,

(13.17)


327

where Γb , Γf ⊂ ∂Ω are closed, Γb ∩ Γf = ∅, and Γb ∪ Γf = ∂Ω. Concerning the heat capacity function m(t) we require t m(t) = m0 + m1 (τ )dτ, t ∈ ⺢, (13.18) 0

1

where m0 > 0 and m1 ∈ L (⺢+ ); cp. the discussion in Section 5.3. We are here interested in the parabolic case, hence the heat conduction function c(t) is supposed to be of the form t c(t) = c0 + c1 (τ )dτ, t ∈ ⺢, (13.19) 0

where c0 > 0 and c1 ∈ L1 (⺢). By a proper rescaling of the time variable we may then assume w.l.o.g. m0 = c0 = 1. ¯ = {θ ∈ C(Ω) ¯ : θ|Γ = 0} be Let Xq = Lq (Ω), 1 ≤ q < ∞, X0 = CΓb (Ω) b equipped with their natural norms and define Aq by means of (Aq θ)(x) = Δθ(x),

x ∈ Ω,

with domains D(Aq ) = D(A0 ) =

¯ : θ|Γ = 0, ∂θ |Γ = 0} , {θ ∈ W 2,q (Ω) b ∂n f ¯ {θ ∈ ∩q>1 D(Aq ) : A0 θ ∈ CΓ (Ω)},

1 < q < ∞,

b

and in X1 define A1 as the closure of e.g. A2 . Let Ω ⊂ ⺢N be bounded and ∂Ω smooth; then it is well known that Aq generates a C0 -semigroup which is analytic in ⺓+ and bounded on each sector Σ(0, φ), φ < π/2. In the Hilbert space case q = 2, the operator A2 is even selfadjoint and negative semidefinite. The spectrum σ(Aq ) is discrete and independent of q, it consists only of simple eigenvalues, (μ − Aq )−1 is compact in Xq , for each μ ∈ ρ(Aq ). Let σ(A) = {−μj }∞ j=0 ,

0 ≤ μ0 < μ1 < . . . ,

(13.20)

and observe that μ0 > 0, unless Γb = ∅, i.e. for the pure Neumann problem. With these notations and with b(t) = e−t e0 (t), (13.17) can be rewritten as v˙ − Aq v = −m1 ∗ v˙ + b ∗ (c1 ∗ Aq v˙ + c1 ∗ Aq v) + f,

(13.21)

where v(t) = θ(t, ·). This problem is of the perturbation form (12.38), with A0 ≡ 0, A1 ≡ Aq , B0 ≡ −m1 , B1 ≡ 0, C0 ≡ c1 Aq , C1 ≡ e0 ∗ c1 Aq . With Yq = XAq , therefore all the assumptions of Corollaries 12.3 and 12.4 are satisfied, and it remains to compute the spectra of (13.21), Λ0 and Σ0 . For this purpose define the functions ˆ 1 (λ)) + (1 + cˆ1 (λ))μj , χj (λ) = λ(1 + m χ∞ (λ) = 1 + cˆ1 (λ), Re λ ≥ 0;

j ∈ ⺞0 , (13.22)

328


since λ ∈ Σ0 iff λ(1 + m ˆ 1 (λ)) − (1 + cˆ1 (λ))Aq is not invertible in B(Yq , Xq ), we see that Σ0 = {λ ∈ ⺓+ : χj (λ) = 0 for some j ∈ ⺞0 ∪ {∞}} (13.23) and Λ0 = {ρ ∈ ⺢ : χj (iρ) = 0 for some j ∈ ⺞0 ∪ {∞}}.

(13.24)

1

Since m1 , c1 ∈ L (⺢)+ by assumption, we have cˆ1 (λ), m ˆ 1 (λ) → 0 as |λ| → ∞, λ ∈ ⺓+ , hence Σ0 as well as Λ0 are compact, and Hq (λ) satisfies |λHq (λ)|B(Xq ) + |Hq (λ)|B(Xq ,Yq ) ≤ M,

λ ∈ ⺓+ , |λ| ≥ R,

(13.25)

where R is sufficiently large. Furthermore, Σ0 ∩ ⺓+ is at most countable, Λ0 has Lebesgue measure 0, and λ0 ∈ Σ0 , Re λ0 > 0, is a cluster point of Σ0 iff χ∞ (λ0 ) = 0. Next we observe that χ∞ (λ) = 0 on ⺓+ if either |c1 | < 1 or c1 is of positive type; cp. Section 5.3. If this is the case and if m1 (t) is nonnegative, nonincreasing, then also χj (λ) = 0 on ⺓+ for j ∈ ⺞, and χ0 (λ) = 0 on ⺓+ \ {0}. Thus except for the pure Neumann case Γb = ∅, in the physically interesting situation we have Σ0 = ∅, hence also Λ0 = ∅. On the other hand, if Γb = ∅ then μ0 = 0, and so Λ0 = Σ0 = {0}. In the latter situation we have Xq = N (Aq ) ⊕ R(Aq ), hence (13.21) splits into an infinite dimensional problem on R(Aq ) with Σ0 = ∅, and into a one-dimensional equation for ϕ(t) = Ω v(t, x)dx, the mean temperature, which reads

ϕ˙ + m1 ∗ ϕ˙ = ψ,

t ∈ ⺢,

(13.26)

where ψ(t) = Ω r(t, x)dx. Returning to the general case, the results of Section 11 and 12 yield e.g. the following properties of (13.21), i.e. (13.17). α,r (a) If Λ ∩ Λ0 = ∅, then the spaces Bp,Λ (Xq ) are spaces of maximal regularity for (13.21), where α ∈ (0, 1), p, r ∈ [1, ∞] (Corollary 12.3). (b) If Λ ∩ Λ0 = ∅, b(t) = e−t e0 (t), H(Xq ) homogeneous, then HΛ (Xq ) is badmissable for (13.21), and there is a Λ-kernel GΛ ∈ L1 (B(Xq )) such that b ∗ GΛ ∈ L1 (B(Xq , Yq )) (Corollary 12.4). (c) If Λ ∩ Λ0 = ∅, f ∈ WΛ+ (X), then v = GΛ ∗ f ∈ WΛ+ (X), σ(v) = σ(f ), exp(v) = exp(f ), and α(ρ; v) = H(iρ)α(ρ; f )

for all ρ ∈ exp (f ).

The same assertions hold for AAPΛ+ (X) instead of WΛ+ (X). If 0 ∈ Λ, f ∈ Cl+ (X) and σ(f ) ⊂ Λ imply v = GΛ ∗ f ∈ Cl+ (X), σ(v) = σ(f ), and v(∞) = H(0)f (∞) (Corollary 11.3 and Theorem 11.2). (d) There is a b-regular resolvent S(t) for the local version of (13.21), and it has the maximal regularity property of type Bpα,r (J; Xq ), and in case 1 < p, q < ∞ also of type Lp (J; Xq ) (Theorems 7.6, 7.5, and 8.7).


329

(e) If Λ0 = ∅, then S(t) = S0 (t) + G(t), (13.21) and its local version are asymptotically equivalent w.r.t. bounded solutions (Theorems 11.3 and 11.4). (f) If Σ0 = ∅, then S is uniformly integrable, and in case 1 < q < ∞, the local version of (13.21) is strongly Lp -stable on the halfline (Theorems 11.3 and 8.8). 13.3 Electrodynamics with Memory Let Ω ⊂ ⺢3 be domain with smooth boundary ∂Ω, which forms a rigid linear isotropic and homogeneous medium. Then Maxwell’s equation are (cp. Section 9.5) Bt + curl E = Ke ,

div B = ρm

Dt − curl H + J = Je ,

div D = ρe

(13.27)

for t ∈ ⺢, x ∈ Ω; here Ke , Je denote the impressed magnetic and electric currents (Ke = 0 in reality), and ρe , ρm the free electric resp. magnetic charges (ρm = 0 in reality). As a boundary condition we impose n × E = 0,

t ∈ ⺢, x ∈ ∂Ω,

(13.28)

where n(x) denotes the outer normal of Ω at x ∈ ∂Ω. Other boundary conditions are of course also of interest, like n × H = 0, but the latter can be treated similar to the approach given below. According to Section 9.5 the constitutive laws are B = dμ ∗ H,

D = dε ∗ E,

J = dσ ∗ E,

where the material functions μ(t), ε(t), σ(t) are of the form t t μ1 (τ )dτ, ε(t) = ε0 + ε1 (τ )dτ, μ(t) = μ0 + 0 0 t σ1 (τ )dτ, t > 0, σ(t) = σ0 +

(13.29)

(13.30)

0

with μ1 , ε1 , σ1 ∈ L1 (⺢+ ) of positive type, σ0 ≥ 0, and μ0 > 0, ε0 > 0 the fundamental constant of the vacuum, c0 = (ε0 μ0 )−1/2 the speed of light in vacuum. In rewriting this equation in the standard form of Section 11, we proceed ˆ1 (λ), Re εˆ1 (λ) ≥ 0 slightly different from Sections 9.5 or 9.6. Since μ0 , ε0 > 0, Re μ on ⺓+ , and μ1 , ε1 ∈ L1 (⺢+ ), by the Wiener-Levy theorem, there are functions η + , η − , ν + , ν − ∈ BV (⺢+ ) such that ±

±

±1/2 ±1/2 (λ) = [dε(λ)] dη , dν (λ) = [dμ(λ)] , ±1/2

±1/2

observe that η ± (0+) = ε0 , ν ± (0) = μ0 ν1± := ν˙ ± ∈ L1 (⺢+ ). The transformations v = dη + ∗ E

,

Re λ ≥ 0;

(13.31)

1,1 , η ± , ν ± ∈ Wloc (⺢+ ) and η1± := η˙ ± ,

w = dν + ∗ H

(13.32)

then induce isomorphisms on any homogeous space, and lead to a particularly nice form of the dynamic equations.

330


In fact, let X = L2 (Ω; ⺢3 )×L2 (Ω; ⺢3 ), u = (v, w)T , and define a linear operator A0 in X by means of v v 0 curl A0 u = A 0 = , u ∈ D(A), (13.33) −curl 0 w w with domain D(A0 ) = {u =

v : v, w ∈ W 1,2 (Ω; ⺢3 ), n × v|∂Ω = 0}; w

and let Y = XA , where A = A0 . It is well-known that A is skewadjoint, hence generates a C0 -group of unitary operators in X. The direct sum decomposition X = R(A) ⊕ N (A) which is also orthogonal, holds, and σ(A) ⊂ i ⺢. If Ω is an exterior domain then σp (A) = {0}, and σc (A) = i⺢ \ {0}. For a bounded domain Ω, the range of A is closed and A|−1 R(A) is compact, hence σ(A) = σp (A) and σp (A) \ {0} consists of eigenvalues ±iξn , 0 < ξn → ∞, as n → ∞, with corresponding eigenvectors forming an orthonormal base of R(A). For a proof of these facts see e.g. the monograph Leis [208], Chap. 8. Ignoring the equations div B = ρm and div D = ρe (these can be considered as the definitions of ρm , ρe ), we obtain by simple manipulations the following reformulation of (13.27), (13.28) and (13.29). u˙ = da ∗ Au + dB ∗ u + f,

t ∈ ⺢,

(13.34)

where da = dη − ∗ dν − , f = (dη − ∗ Je , dν − ∗ Ke )T , and the operator-valued kernel B ∈ BV (⺢+ ; B(X) ∩ B(Y )) is given by −dη − ∗ dη − ∗ σ(t) 0 B(t) = . (13.35) 0 0 Along with the problem (13.34) on the line we also consider the corresponding problem on the halfline z˙ = da ∗ Az + dB ∗ z + f,

t ≥ 0, z(0) = z0 .

(13.36)

We are now in position to apply the theory developed in this book to the study of (13.34) and (13.36); observe that these problems are of hyperbolic type. There are different lines of approach to the problem in question; we use here that based on subordination and perturbation, which yields the strongest result. However, we have to assume that μ, ε are bounded Bernstein functions. (a) As in Section 9.5, a(t) is completely monotonic by Lemma 4.3. The characteristic numbers κ, α, ω for a(t) are 1 ε0 μ0 κ = 1/c0 , ω = 0, α = (μ1 (0+) + ε1 (0+) ), (13.37) 2 μ0 ε0


331

and for the case σ ≡ 0, by Theorem 4.2 and 4.4, the resolvent S0 (t) for (13.36) with σ ≡ 0 is S0 (t) = ec0 At e−αc0 t + S1 (t), t ≥ 0, (13.38) where eAt denotes the C0 -group generated by A and S1 ∈ C((0, ∞); B(X)) is bounded, S1 is also B(X)-continuous at t = 0 and S1 (0) = 0 in case α < ∞. Observe that S0 (t) is B(X)-continuous on ⺢+ iff α = ∞, i.e. if μ1 (0+) = ∞ or ε1 (0+) = ∞. The general case is now a consequence of Theorem 6.1 in the form of Corollary 6.2, to the result that (13.36) admits an exponentially bounded e−t -regular resolvent S(t). (b) Next we may apply Theorem 12.2, assuming again σ = 0 first. With b(t) ≡ 1, ωb = β = 0, this result yields a Λ-kernel GΛ (t) for (13.34) whenever α > 0 and Λ ∩ Λ0 = ∅. GΛ (t) is of the form GΛ (t) = ec0 At e−αc0 t e0 (t) + G0 (t),

t ∈ ⺢,

(13.39)

with G0 ∈ L1 (⺢+ ; B(X)) ∩ L∞ (⺢; B(X)) B(X)-continuous on ⺢, GΛ (t) → 0 in B(X) as |t| → ∞, and the jump relation lim GΛ (t) − lim GΛ (t) = I,

t→0+

t→0−

holds in the strong sense. Moreover, each homogeneous space HΛ (X) is e−t admissable if Λ ∩ Λ0 = ∅. Observe that Λ0 is compact anyway, and that GΛ is integrable, not just strongly integrable, and even uniformly integrable in case α = ∞. Via perturbation results these properties remain valid for σ = 0; this is clear for σ0 = 0, by Theorem 12.1, but also true in case σ0 > 0. For the latter one has to modify A by incorporating the damping term −σ0 ε0 /μ0 in the left upper corner of A defined by (13.33). eAt is then no longer unitary, but is still a bounded C0 -semigroup, which respects the decomposition X = R(A) ⊕ N (A). (c) It remains to compute the spectra Λ0 and Σ0 of (13.34) and (13.36). For this purpose observe that the equation (λ − da(λ)A − dB(λ))u =f with u = (v, w)T and f = (g, h)T reduces to the boundary value problem ϕ(λ)v − curl w n×v

= dk(λ)g

in Ω,

= 0 on ∂Ω, = dk(λ)h in Ω,

(13.40)

ψ(λ)w + curl v = 1/da = dε dμ, ψ = λdk, and ϕ = ψ + dk dσ/ dε. Taking the inner where dk product with v in the first equation of (13.40), with w in the third, addition and an integration by parts yield v) + dk(h, w). ϕ|v|22 + ψ|w|22 = dk(g,

(13.41)

332


Since k ∈ BV (⺢+ ), this identity implies an estimate of the form |v|22 + |w|22 ≤ C(λ)2 (|g|22 + |h|22 ) for all v, w ∈ L 2 (Ω; ⺢3 ), provided ϕψ ∈ (−∞, 0]. Therefore we obtain Σ0 ⊂ {λ ∈ ⺓+ : ϕ(λ)ψ(λ) ∈ (−∞, 0]}. However, the definitions of ϕ and ψ imply dε + dσ); ϕψ = (λdμ)(λ > 0 and Re (λdε) > 0 for since ε, μ are Bernstein functions we have Re (λdμ) Re λ > 0, and so the assumption that σ1 is of positive type shows Σ0 = iΛ0 , i.e. ϕψ ∈ (−∞, 0] can only happen for λ ∈ i⺢. But then for λ = iρ = 0 we must have −ρ Im μ 1 (iρ) = 0, −ρ Im ε1 (iρ) = 0,

σ0 = 0, Re σ 1 (iρ) = 0;

since μ1 , ε1 are completely monotonic this is only possible in case μ1 ≡ ε1 ≡ 0, which contradicts the assumption α > 0 made above. Thus we conclude Σ0 = Λ0 ⊂ {0}. (d) Let us consider the point λ = 0 separately. If the domain Ω is not bounded then R(A) will in general not be closed, hence there is no hope for Λ0 = ∅, without changing the topology, at least. On the other hand, if Ω is bounded then R(A) is closed; hence we may split the problem into one in R(A) and one in N (A), since B(t) respects the orthogonal decomposition X = R(A) ⊕ N (A). In N (A) we obtain an ordinary system of Volterra equations given by v˙ + dη − ∗ dη − ∗ dσ ∗ v = g,

w˙ = h;

(13.42)

if σ(∞) > 0 the first of these equations can be solved in any homogeneous function space H(⺓3 ), while the second (both if σ(∞) = 0) depends on the invariance of H(⺓3 ) w.r.t. integration. The problem restricted to R(A) has the property Λ0 = Σ0 = ∅; recall we are assuming Ω bounded, here. In fact, since A|−1 R(A) is compact, either (A + B(∞))|R(A) is invertible, i.e. 0 ∈ Λ0 , or 0 is an eigenvalue of (A + B(∞))|R(A) . To exclude the latter assume u ∈ R(A) and Au + B(∞)u = 0; with u = (v, w)T this implies curl w = γv , curl v = 0, n × v|∂Ω = 0, where γ = σ(∞) μ(∞)/ε(∞). If γ = 0 then obviously u ∈ N (A), hence u = 0. If γ > 0 then 0 = ( curl v, w) = (v, curl w) = γ|v|2 , i.e. v = 0, and therefore curl w = 0 which means u ∈ N (A) and so again u = 0. This shows 0 ∈ Λ0 , hence Λ0 = Σ0 = ∅, when (13.34) and (13.36) are restricted to R(A). (e) To conclude the discussion, let Ω be bounded (hence R(A) closed) and simply connected, α > 0 (which excludes ε(t) ≡ ε0 , μ(t) ≡ μ0 ). Then the resolvent


333

S(t) is integrable on R(A), even uniformly if α = −∞, every homogeneous space H(R(A)) is e−t -admissible for (13.34), and (13.34) and (13.36) are asymptotically equivalent, by Theorems 11.3 and 11.4. If in addition σ(∞) > 0, the first equation in (13.42) admits also an integrable resolvent. Since there are no magnetic currents in reality, we actually have h = 0, and so the resolvent S(t) is even integrable on X0 = {u ∈ X : div w = 0, w · n|∂Ω = 0} the space of physical interest. Thus the presence of a memory in at least one of the material functions μ(t) or ε(t) stabilizes the high frequencies; while for low frequencies a stationary resistance 1/σ(∞) < ∞ does this job no matter how large ε1 , μ1 and σ are. 13.4 Ergodic Theory A question very much related to the subject of integrability of resolvents S(t) for

t

u(t) = f (t) + 0

or

a(t − τ )Au(τ )dτ,

t

u(t) = f (t) + 0

A(t − τ )u(τ )dτ,

t ≥ 0,

(13.43)

t ≥ 0,

(13.44)

is that of existence of P = limt→∞ S(t) in various senses, and what the nature of this limit will be. More precisely, the resolvent S(t) and (13.43) resp. (13.44) are called uniformly resp. strongly resp. weakly ergodic if P = limt→∞ S(t) exists in the uniform resp. strong resp. weak operator topology. This type of ergodicity will be denoted by (j, E)-ergodicity, where j denotes any of the symbols u, s, w representing tuniformly, strongly, weakly, respectively. If instead only the Cesaromeans t−1 0 S(τ )dτ converge to P in one of these operator topologies, then S(t) and (13.43) resp. (13.44) will be called uniformly resp. strongly resp. weakly Cesaro ergodic, and this ∞type will be denoted by (j, C)-ergodicity. If merely the Abel-means λH(λ) = 0 [λe−tλ ]S(t)dt converge as λ → 0 then the term Abelergodicity or more precisely (j, A)-ergodicity will be used. Observe that the limit P is unique and the following implication scheme is valid (u, E) ⇓ (u, C) ⇓ (u, A)

⇒ ⇒ ⇒

(s, E) ⇒ ⇓ (s, C) ⇒ ⇓ (s, A) ⇒

(w, E) ⇓ (w, C) ⇓ (w, A).

Here the horizontal implications are obvious, while the vertical arrows are implied by so-called Abelian theorems. In general, none of these implications can be reversed; a result asserting the converse of a vertical arrow is called a Tauberian theorem, in honour of A. Tauber who was the first to use this type of argument in the case of numerical power series; see Tauber [323]. The recent paper by Arendt and Pr¨ uss [14] contains an account of the Tauberian theory for the vector-valued Laplace transform; see also Hille and Phillips [180], Chapter XVIII, Arendt and

334


Batty [11], [12], and Batty [20]. For the classical Tauberian theory we refer to the monographs of Doetsch [100] and Widder [339], [340]. Let us briefly discuss ergodic theory in the semigroup case a(t) ≡ 1; see Hille and Phillips [180]. If ω0 (A) ≤ 0 (the type of eAt ) then (j, A)-ergodicity can be completely characterized. eAt is (w, A)-ergodic iff it is (s, A)-ergodic iff λH(λ) is bounded as λ → 0+, and N (A)⊥ ∩ N (A∗ ) = {0}; eAt is (u, A)-ergodic iff it is (w, A)-ergodic and R(A) is closed. Note that N (A)⊥ ∩ N (A∗ ) = {0} is automatically satisfied in case X is sun-reflexive w.r.t. eAt , in particular if X itself is reflexive, in virtue of de Pagter’s characterization of sun-reflexivity; see de Pagter [80]. The Abelian ergodic limit P if it exists has a very simple structure, namely it is the projection onto N (A) along R(A). By means of vector-valued real Tauberian theorems, eAt is (s, C)-ergodic if it is (w, A)-ergodic and eAt is bounded, and it is (u, C)-ergodic if it is (u, A)-ergodic and eAt is bounded and eventually B(X)-continuous. Furthermore, (j, A)-ergodicity and boundedness of |eAt | + t|AeAt | imply (j, E)-ergodicity, since in this case eAt is feebly oscillating; note that bounded analytic semigroups are covered by this result. Another very interesting theorem was recently discovered independently by Arendt and Batty [11] and Lyubich and Phong [229]; it states that (s, A)-ergodicity, boundedness of eAt , σ(A) countable, and σp (A∗ )∩i⺢ ⊂ {0} imply (s, E)-ergodicity. This result has been used e.g. by Wyler [341] to obtain asymptotic stability of several classes of dissipative wave equations. To conclude this discussion, observe that integrability of eAt is equivalent to ω0 (A) < 0, in which case eAt is (u, E)-ergodic with ergodic limit P = 0. In the Hilbert space case, by Gearhart’s theorem the same result is valid for strongly integrable semigroups; cp. Section 12.7. Returning to evolutionary integral equations, let us first observe that in any of the integrability results for the resolvent proved in Sections 10 and 12, S(t) → 0 in B(X), except for Section 10.4 where the convergence still takes place in the strong operator topology. But although, contrary to the semigroup case, integrability of S(t) can in general not be expected to imply (s, E)-ergodicity, the situation is much more interesting if the resolvent is not integrable. To understand this, let us consider linear isotropic incompressible viscoelasticity once more. If as in Section 5.6, A denotes the Stokes operator in L20 (Ω; ⺢3 ) and a(t) is a creep function with log a1 (t) convex, then Corollary 10.3 states that S(t) is integrable iff a(t) ≡ a∞ t and 0 ∈ ρ(A). The latter is the case if the underlying domain Ω ⊂ ⺢3 is bounded, and it is also true for some types of unbounded domains, however, not for all. In Arendt and Pr¨ uss [14] it has been shown that one always has S(t) → 0 as t → ∞ in the strong sense, provided a(t) ≡ a∞ t, i.e. the material is not purely elastic. This result as well as others follow from the main result of the paper just quoted, which applies to (13.43). Theorem 13.1 Suppose (13.43) admits a resolvent S(t), bounded on ⺢+ , let a(t) be of subexponential growth and such that a(λ) admits continuous extension in ⺓ ∪ {∞} to ⺓+ ; assume that P = limλ→0+ λH(λ) exists in the strong sense, and define ρ(a, A) = {iη ∈ i⺢ : there is ε > 0 such that [(1 − a(λ)A)−1 − P ]/λ (13.45)


335

admits a strongly continuous extension to ⺓+ ∪ i[η − ε, η + ε]}. In addition suppose the following conditions are satisfied. (E1) The singular set iE := i⺢ \ ρ(a; A) is countable, and 0 = μ ∈ E, a(iμ) = 0, ∞ 0 = μ ∈ E, a(iμ) = ∞

⇒ ⇒

R(1 − a(iμ)A) = X; N (A)⊥ ∩ N (A∗ ) = {0}.

(E2) For all μ ∈ E there exists a constant C(μ) > 0 such that for all x ∈ D(A) |

0

t

e−iμs (a ∗ S(s) − a(iμ)S(s)Ax)ds| ≤ C(μ)|x|A |

t 0

e−iμs S(s)Axds| ≤ C(μ)|x|A

if a(iμ) ∈ ⺓, and

if a(iμ) = ∞.

˙ (E3) There exist τ ≥ 0, M ≥ 0 such that |S(t)x| ≤ M |x|A , for all x ∈ D(A), t ≥ τ. Then limt→∞ S(t)x = P x for all x ∈ X, where P = (1 − a(0)A)−1 if a(0) ∈ ⺓, and P is the projection onto N (A) along R(A) if a(0) = ∞. This result contains as a special case the convergence theorem due to Arendt and Batty, and Lyubich and Phong which was mentioned above; take a(t) ≡ 1. Although the conditions seem to be difficult, a check can be made in many specific instances, including the application to viscoelasticity discussed above; see Arendt and Pr¨ uss [14]. As an illustrative example, suppose that A is an m-dissipative operator in a Hilbert space X such that ρ(A) ⊃ i⺢, and consider a(t) = cos t; then Theorem 13.1 implies S(t) → I strongly as t → ∞. In the paper just quoted one also finds complete characterizations of Abel ergodicity of S(t) for equations of scalar type. Together with boundedness of S(t) on ⺢+ these imply Cesaro ergodicity, i.e. yield the mean ergodic theorem for resolvents. A different mean ergodic theorem has been recently obtained in Lizama [217]. There are yet no such studies for nonscalar problems (13.44), nor are there results on asymptotic almost periodicity of resolvents like the theorem of Jacobs, Glicksberg and Deleuuw for C0 -semigroups; cp. Section 11.4. 13.5 Semilinear Equations For about three decades semilinear abstract Cauchy problems of the form u(t) ˙ = Au(t) + F (t, u(t)), u(0) = u0 ,

t ≥ 0,

(13.46)

in a Banach space X have been studied in many different settings. Here A denotes the generator of a C0 -semigroup T (t) in X and F a nonlinear function which is considered as a perturbation of the main term Au. We refer to the monographs Friedman [122], Goldstein [133], Haraux [172], Henry [178], Martin [235], Pazy

336


[267], Reed [287], Tanabe [319], and Webb [338] for details of the approaches explained briefly below, as well as for the original literature. The basic idea for this type of approach is to consider the integral equation u(t) = T (t)u0 +

t 0

T (t − s)F (s, u(s))ds,

t ≥ 0,

(13.47)

which follows from the variation of parameters formula for the linear Cauchy problem u˙ = Au + f , u(0) = u0 . In this formulation the unbounded operator A does only appear in terms of the semigroup T (t), which makes e.g. the application of fixed point theorems for the solution of the integral equation possible. A typical argument uses the contraction mapping principle in the space Z = C(J; X), where J = [0, τ ]. Suppose the nonlinearity F : J × X → X is continuous and globally Lipschitz, i.e. |F (t, u) − F (t, v)| ≤ L|u − v|,

for all u, v ∈ X, t ∈ J;

then the map K : Z → Z defined by the right hand side of (13.47) is well-defined and easily seen to be a contraction w.r.t the norm ||u|| = sup{|u(t)|e−ωt : t ∈ J},

u ∈ Z,

provided ω is chosen large enough. Consequently, (13.47) has a unique continuous solution u(t), for every u0 ∈ X, and the map u0 → u(·) is Lipschitz from X to Z. Solutions of the integral equation (13.47) are called mild solutions of (13.46). This argument has been refined in many directions, e.g. localizing the Lipschitz condition, or replacing it by compactness assumptions involving measures of noncompactness and applying fixed point theorems for set contractions, or by allowing A to depend on t. This way also a qualitative theory for mild solutions of (13.46) can be developed which to a large extent parallels that of ordinary differential equations in finite dimensions. If (13.46) is parabolic in the sense that the semigroup T (t) generated by A is analytic (and w.o.l.g. bounded on ⺢+ ), the assumptions on F can be relaxed considerably, due to the regularizing effect of the convolution with T (t). In this situation it is enough to assume that F : ⺢+ × Xα → X is continuous and e.g. Lipschitz in its second argument, where Xα = D((−A)α ) equipped with its graph norm, and α ∈ [0, 1). In the parabolic case one can go a step further, utilizing the available maximal regularity results. In fact, by means of maximal regularity of type C β , the choice α = 1 is possible, if in addition F is Hölder continuous of degree β > 0 w.r.t. its first argument, the compatibility condition Au0 +F (0, u0 ) ∈ DA (β, ∞) is satisfied, and Fu (0, u0 ) = 0 holds for the Fréchet-derivative Fu of F w.r.t. u. The solutions obtained this way will even be strong solutions and enjoy the extra regularity u ∈ C β (J; XA ), u˙ ∈ C β (J; X). Being only based on the variation of parameters formula, this approach can almost directly be generalized to semilinear evolutionary integral equations of the

13. Applications and Complements form

t

u(t) ˙ = 0

337

dA(τ )u(t − τ ) + V (u)(t), u(0) = u0 ,

t ≥ 0,

(13.48)

where A ∈ BVloc (⺢+ ; B(Y, X)) is such that the corresponding linear problem u = A ∗ u + f admits a weak resolvent S(t) in X, and V : C(⺢+ ; X) → X is a Volterra operator on the halfline in the sense that V (u)(t) depends only on the values {u(s) : s ∈ [0, t]}, and satisfies appropriate smoothness or compactness conditions. The integral equation corresponding to (13.48) is of course t u(t) = S(t)u0 + S(t − τ )V (u)(τ )ds, t ≥ 0. (13.49) 0

Examples for possible nonlinearities V which fit into this framework are e.g. Volterra operators like t k(t, s, u(s))ds). V (u)(t) = F (t, u(t), 0

Depending on the regularity properties of the resolvent S(t) for the linear part of (13.48) and those of V , the solvability behaviour of (13.48) can be studied as indicated above for the pure abstract semilinear Cauchy problem (13.46). Similarly, the equation on the line ∞ u(t) ˙ = dA(τ )u(t − τ ) + V (u)(t), t ∈ ⺢, (13.50) 0

can be treated, where now A ∈ BV (⺢+ ; B(Y, X)), the corresponding linear equation on the line u˙ = dA ∗ u + f admits a ⺢-kernel in the sense of Section 11.5, and V is a nonlinear Volterra operator on the line, i.e. V (u)(t) depends only on the values {u(s) : s ≤ t}, the history of u at time t. The equivalent integral equation is then given by t

u(t) = −∞

G(t − τ )V (u)(τ )ds,

t ∈ ⺢.

(13.51)

By means of this reformulation, based on the theory of reolvents presented in this book, also a qualitative theory for (13.48) and (13.50) can be developed, in particular stability results, saddle point properties, Hopf bifurcation, etc. can be obtained as in the case of (13.46). There is a already a number of papers available where these ideas have been carried out in various settings, hyperbolic as well as parabolic. Mostly the cases A(t) ≡ A, i.e. the resolvent is a C0 -semigroup, and A(t) ≡ tA, i.e. S(t) is a cosine family have been treated so far; see e.g. Aizicovici [4], Da Prato and Lunardi [72], Engler [109], [111], Heard [174], [175], [176], Heard and Rankin [177], Lunardi [224], Lunardi and Sinestrari [228], [227], Milota [247], [248], Milota and Petzeltov´ a [250], [249], Petzeltov´ a [268], Sinestrari [306], Tesei [324], Travis and Webb [327], Webb [336], [337], Yamada [344], Yamada and Niikura [345], Yoshida [346]. For the general case, however, much work is left to be done.

338


13.6 Semigroup Approaches In this subsection some of the meanwhile numerous different semigroup approaches to evolutionary integral equations will be presented and discussed. Since the work of Hale [160] on functional differential equations in finite dimensions these methods have become quite popular and have been employed by many people; see Gripenberg, Londen and Staffans [156] for the finite-dimensional case. For evolutionary integral equations they go back to Miller [243], Grimmer and Miller [142], [143], Miller and Wheeler [245], and have been further developed and extended by Chen and Grimmer [40], [41], Desch and Grimmer [82], [83], Desch, Grimmer and Schappacher [86], Desch and Schappacher [92], [93], Di Blasio, Kunisch and Sinestrari [96], [97], Grimmer [139], Grimmer and Schappacher [147], Grimmer and Zeman [148], Kunisch and Schappacher [202], Jeong [186], Nakagiri [256], Navarro [257], Sinestrari [307], Slemrod [308], [309], Staffans [315], and Tanabe [322]; see also Renardy, Hrusa, and Nohel [289], Chapter V. This list does not claim any completeness. a) To illustrate the basic ideas, let us consider an equation with main part of the form t B(t − s)u(s)ds + f (t), u(0) = x, t ≥ 0, (13.52) u(t) ˙ = Au(t) + 0

in a Banach space X, where A is a closed linear densely defined operator in X, and B ∈ L1loc (⺢+ ; B(XA , X)); the data should satisfy x ∈ X and f ∈ L1loc (⺢+ ; X) for the moment. We want to rewrite (13.52) as an abstract Cauchy problem of the form z(t) ˙ = Az(t), z(0) = z0 , t ≥ 0, (13.53) in an appropriate Banach space Z. There are many different ways how this can be done. The so-called forcing function approach proceeds as follows. Choose a function space F (⺢+ ; X) where the forcing functions f one is interested in should belong to, e.g. F = Cub , or F = Lp , etc., and let Z = X × F (⺢+ ; X). For a given strong solution u(t) of (13.52) define

t

z(t) = (u(t), g(t)), g(t)(r) = f (t + r) + 0

B(t + r − s)u(s)ds,

t, r ≥ 0; (13.54)

in particular, z(0) = z0 = (u(0), g(0)) = (x, f ). It is then straight forward to compute that, formally, z(t) satisfies (13.53), with A defined by A δ0 A= , (13.55) B(·) D where δ0 f = f (0), B(·)x(r) = B(r)x, and Df (r) = f (r). The natural domain of definition for A is of course D(A) = D(A) × D(D),


339

as soon as B(·)x ∈ F(⺢+ ; X) for each x ∈ D(A). For the solutions u(t) of (13.52) we have the representation via the variation of parameters formula t u(t) = S(t)x + S(t − τ )f (τ )dτ, t ≥ 0, 0

where S(t) denotes the resolvent of (13.52). On the other hand, the solution of (13.53) is given by z(t) = eAt z0 t ≥ 0; this leads to the following formal representation of eAt in terms of S(t) and B(t). S(t) [S∗](t) At t e = , t ≥ 0. (13.56) B(t − τ + ·)S(τ )dτ Tt + [R(·)∗](t) 0 Here (Tt f )(r) = f (t +r) denotes the translation semigroup, [S∗](t)f = (S ∗ f )(t) is t a funny notation for the convolution, and R(r)(t) = 0 B(r + t − τ )S(τ )dτ . Thus the first component of z(t) = eAt z0 is nothing else than the variation of parameters formula. Since S(t) appears in the left upper corner of eAt , well-posedness of (13.53) implies existence of the resolvent S(t) for (13.52), no matter which function space F has been chosen. Therefore whenever it can be proved that A generates a C0 semigroup in Z then well-posedness of (13.52) follows, a very appealing aspect of this approach. The converse of this, however, is in general not true since it is a priori not clear that the entries in the second row of eAt are well-defined; note that B(t) is only defined on XA . For the same reason A need not even be closable, at least without further assumptions, which is clearly a drawback of this method. In general, it also seems to be very difficult to apply directly the Hille-Yosida theorem on generation of C0 -semigroups, well-posedness of (13.53) is usually obtained via perturbation methods. A typical result requires A to be a generator; then the operator A0 defined by the diagonal part of A is again a generator at least if the translation semigroup is strongly continuous in F . Therefore it is natural to consider the off-diagonal part of A as a perturbation B of A0 , and to apply one of the many perturbation results for semigroups. This works e.g. in case F (⺢+ ; X)) ⊂ C(⺢+ ; X) and B(·)x ∈ D(D) for all x ∈ D(A). Another possibility is to consider the case B ≡ 0 first; the representation (13.56) shows that then A is a generator in Z whenever A is a generator in X, the translation semigroup is strongly continuous in F (⺢+ ; X), and this space is continuously embedded into L1loc (⺢+ ; X). A perturbation result due to Desch and Schappacher [92] then implies that (13.53) is well-posed if B(·)x ∈ D(D), for each x ∈ D(A); here D(D) denotes the generalized domain of D. Note, however, that the semigroups obtained via these methods will never be analytic nor compact, this has nothing to do with the corresponding properties of S(t). Therefore maximal regularity results for (13.52) cannot be proved this way. The same remark applies to stability questions; even if S(t) is integrable, the type of eAt cannot be negative unless the type of S(t) already is; this implies that

340


it is not possible to obtain sharp integrability results for the resolvent like those obtained in Section 10 by means of this approach. b) Another semigroup formulation which has been developed, in particular for the parabolic case is the history function approach. To explain it in some detail, consider the problem t u(t) ˙ = Au(t) + B(t − s)u(s)ds, t ≥ 0, (13.57) −∞

u(0) = x, u(t) = φ(t),

t < 0,

where X, A, and B are as before. Fix a function space F (⺢− ; X) serving as the space where the history functions φ belong to, and let Z = X × F(⺢− ; X); the initial value for the Cauchy problem will be z0 = (x, φ). If u(t), t ∈ ⺢, denotes the solutions of (13.57) then z(t) = (u(t), u(t + ·)) ∈ Z should be the solution of (13.53), where A still has to be determined. For the infinite memory problem (13.57) the variation of parameters formula becomes

t

u(t) = S(t)x + 0

S(t − r)

0 −∞

B(r − τ )φ(τ )dτ dr,

t ≥ 0;

therefore a formal computation shows that A is given by A < B(·), · > A= , (13.58) 0 D ∞ where < B(·), φ > = 0 B(s)φ(−s)ds. Here, however, the domain of A is restricted by the requirement φ(0) = x. Proceeding as before, consider the case B ≡ 0 first, and let A0 denote the corresponding operator in Z; if A is a generator in X and F is such that the translation semigroup is strongly continuous then A0 is a generator on D(A0 ), where D(A0 ) = {z = (x, φ) ∈ D(A) × D(D) : φ(0) = x}. In fact, the action of the semigroup is given by

A(t+s) e x for t + s > 0 A0 t At e (x, φ) = (e x, ). φ(t + s) for t + s < 0

(13.59)

But now there is a problem with the perturbation term < B(·), φ >; namely this term is not defined at all unless B(s) is bounded in X, the trivial case. One might think that a way out would be to consider history functions φ with values in XA only; but then the problem with B ≡ 0 is not well-posed, as the representation (13.59) shows. One possibilty to overcome these difficulties is to consider kernels B(t) of the form B(t) = AL(t) + K(t), t > 0, with L(t), K(t) ∈ B(X).


341

Then the perturbation term coming from K leads to a bounded perturbation of A0 -at least under appropriate regularity assumptions- hence causes no problems; so we may assume K ≡ 0 for simplicity. But then A is of the form A = A0 (I + L), where I denotes the identity in Z, and ∞ L(x, φ) = ( L(s)φ(−s)ds, 0), (x, φ) ∈ Z. 0

It is easy to check that I + L is invertible, therefore A is a generator iff (I + L)A0 is such; but ∞ ∞ ˙ L(s)φ (−s)ds, 0) = (L(0)x + LA0 (x, φ) = ( L(s)φ(−s)ds, 0), 0

0

for all (x, φ) ∈ D(A0 ), shows that LA0 is a bounded perturbation, provided L˙ satisfies appropriate regularity assumptions. The domain of A will then be ∞ L(s)φ(−s)ds ∈ D(A), φ ∈ D(D), φ(0) = x.} D(A) = {z = (x, φ) ∈ Z : x + 0

Observe that strong solutions of (13.53) will only be mild solutions of (13.52), in general, since the first component of z(t) does not necessarily belong to D(A). c) Since many people still believe that it is useful to have a semigroup formulation of evolutionary integral equations let us show by means of results proved in this book, how semigroups can be constructed. For this purpose consider the infinite memory problem t dB(t − s)u(s)ds, t ≥ 0, (13.60) u(t) ˙ = Au(t) + −∞

u(0) = x, u(t) = φ(t),

t < 0,

where A generates an analytic C0 -semigroup in X, and B ∈ BV (⺢+ ; B(XA , X)) with B(0) = B(0+) = 0. As a phase space we choose e.g. Z = {(x, φ) ∈ XA × C α (⺢+ ; XA ) : y(x, φ) ∈ DA (α, ∞), φ(0) = x},

where

∞

dB(s)φ(−s)ds.

y(x, φ) = Ax + 0

Then for this situation the maximal regularity result Theorem 7.4 with S0 (t) = eAt shows that the solution u(t) of (13.60) belongs to C α (−∞, T ]; XA ), whenever (x, φ) ∈ Z. For this one has to observe that u(t) can be written as u = x + 1 ∗ Sy + S ∗ f,

for some f ∈ C0α (⺢+ ; X),

and 1 ∗ Sy ∈ C0α (⺢+ ; XA ) by the compatibility condition. It is not difficult to see that the latter, which may be written as u(t) ˙ ∈ DA (α, ∞), is also preserved, and therefore the map (x, φ) → (u(t), u(t + ·)) is a strongly continuous semigroup in Z.

342


The maximal regularity results from Sections 7 and 8 yield many other choices of phase spaces Z for which the semigroup can be constructed. d) Besides the semigroup approaches discussed above there are many other possibilities, especially for concrete partial differential-integral equations like those arising in viscoelasticity, for example. This is of course due to the additional structure such problems enjoy, and which to a large part is abandoned when formulating them abstractly. For semigroups constructed in this spirit the interested reader is referred to the papers of Slemrod [308], [309], Navarro [257], and to the more recent ones of Desch and Grimmer [83], Desch and Miller [90], Leugering [212], and Staffans [315]. 13.7 Nonlinear Equations with Accretive Operators Since the early 1970s Volterra equations of scalar type have been under consideration also in the nonlinear case. More precisely, equations of the form t a(t − s)Au(s)ds f (t), t ≥ 0, (13.61) u(t) + 0

have been studied in general Banach spaces, where a ∈ L1loc (⺢+ ), f ∈ L1loc (⺢+ ; X), and A is a possibly nonlinear multivalued m-accretive operator in X. Recall that an operator A : X → 2X is called accretive if |x1 − x2 | ≤ |x1 − x2 + μ(y1 − y2 )|,

for all yj ∈ Axj , μ > 0,

(13.62)

is satisfied; A is called m-accretive if it is accretive and in addition R(I + μA) = X holds for some μ > 0. The latter means that for each y ∈ X there is x ∈ D(A) = {x ∈ X : Ax = ∅} such that μ−1 (y − x) ∈ Ax. For the theory of accretive and m-accretive operators and nonlinear semigroup theory we refer e.g. to the monograph Deimling [81]; note that ‘m-accretive’ is called ‘hyperaccretive’ there. A function u ∈ L1loc (⺢+ ; X) is called a strong solution of (13.61) if u(t) ∈ D(A) for a.a. t ≥ 0, and there is a function w ∈ L1loc (⺢+ ; X) with w(t) ∈ Au(t) a.e., and u + a ∗ w = f a.e. on ⺢+ . u ∈ L1loc (⺢+ ; X) is called a mild solution of (13.61) if there are functions fn → f in L1loc (⺢+ ; X), and strong solutions un of (13.61) with f replaced by fn , such that un → u in L1loc (⺢+ ; X). Beginning with the papers McCamy [230], Barbu [17], and Londen [218], (13.61) for nonlinear A has been studied in many different settings under various assumptions by means of different techniques. The main emphasis has been on existence of solutions and well-posedness questions, and on the asymptotic behaviour of the solutions. In the papers Barbu [17] and Londen [218] it is assumed that X is a Hilbert space, and that A = ∂φ is the subdifferential of a lower semi-continuous (l.s.c.) proper convex functional; see e.g. Deimling [81] for the definition and properties of such functionals. The nonlinear operator A was then replaced by its Yosida approximation Aμ defined by Aμ = (I − Jμ )/μ, Jμ = (I + A)−1 ,

μ > 0,

(13.63)


343

which is globally Lipschitz; hence there are unique solutions uμ of the approximating equations t uμ (t) + a(t − s)Aμ uμ (s)ds f (t), t ≥ 0. (13.64) 0

For 0 = a ∈ BV (⺢+ ) 1-monotone and of positive type, Barbu obtained bounds on uμ and Aμ uμ which allowed for passing to the limit μ → 0+, thus obtaining strong solutions, and some further asymptotic properties in case a(∞) > 0. Instead, Londen assumed a, a˙ ∈ BVloc (⺢+ ) and a(0) > 0 to obtain essentially the same conclusion; Gripenberg [150] extended these results to general maximal monotone operators A in a Hilbert space X; see also Londen [219]. For a further development of the asymptotic theory of (13.61) in this setting we refer to Clément, McCamy, and Nohel [48]. An essential step forward was made in the paper by Crandall and Nohel [62]. They consider the perturbed nonlinear Cauchy problem u˙ + Au G(u), u(0) = x,

t ∈ J = [0, T ],

(13.65)

where A is m-accretive in the Banach space X and G : C(J; D(A)) → L1 (J; X) is a functional operator subject to the Lipschitz estimate t |G(u) − G(v)|L1 ((0,t);X) ≤ γ(s)|u − v|L∞ ((0,s);X) ds, t ∈ J. 0

They show the existence, uniqueness and continuous dependence of (continuous) integral solutions for (13.65), which are even strong solutions if x ∈ D(A), X is reflexive, and R(G) ⊂ BV (J; X). Their method is basically a perturbation argument, it relies on the Crandall-Liggett theorem of nonlinear semigroup theory (see e.g. Deimling [81]) and on the contraction mapping principle. Observe that (13.61) can be reduced to (13.65) by inverting the convolution with a, provided there are k0 > 0, k1 ∈ BV (J) such that k0 a + k1 ∗ a = 1, which means that a, a˙ ∈ BV (J), a(0+) > 0, i.e. Londen’s condition. Even more, the approximate solutions uμ converge uniformly in X to u as μ → 0+. Observe that all results mentioned so far require 0 < a(0+) < ∞. Clément and Nohel [50] discovered that kernels a(t) which later on have been called completely positive play an important role also in the nonlinear case. Specifically they proved the positivity (w.r.t. a cone P ) of mild solutions of (13.61) which are obtained as weak limits of the uμ , whenever (I + μA)−1 P ⊂ P for each μ > 0, a(t) is a completely positive function, and f is of the form f = x + a ∗ g, with x ∈ P and g(t) ∈ P for a.a. t. In Clément [43] the important inequality t |u1 (t) − u2 (t)| ≤ |x1 − x2 | + a(t − s)|g1 (s) − g2 (s)|ds, t > 0, (13.66) 0

is proved for mild solutions uj of (13.61) with f replaced by fj = xj + a ∗ gj , which are limits of the approximate solutions uμ,j . This estimate is the basis on the

344


results on asymptotic behaviour of mild solutions obtained in Clément [43], and Clément and Nohel [51]. The asymptotic theory was carried on by Aizicovici [5], Baillon and Clément [15], N. Hirano [181], N. Kato, K. Kobayasi, and I. Miyadera [192], and N. Kato [190], [191]. See also Clément and Mitidieri [49] for other qualitative properties of (13.61) like invariance, monotonicity, and comparison principles, whenever a is completely positive and A m-accretive. Let a(t) be completely positive and k(t) the creep function associated with a, as in Section 4. If f is of the form f = x + a ∗ g, x ∈ X and g ∈ L1loc (⺢+ ; X), then (13.61) is equivalent to t d [k0 u(t) + k1 (t − s)u(s)ds] + k∞ u(t) + Au(t) (k1 (t) + k∞ )x + g(t), dt 0 k0 u(0) + (k1 ∗ u)(0) = k0 x, t ∈ ⺢+ . (13.67) Since the linear Volterra operator L defined by (Lu)(t) =

d [k0 u(t) + dt

t 0

k1 (t − s)u(s)ds] + k∞ u(t),

t ∈ ⺢+ ,

is m-accretive in Lp (⺢+ ; X) (see also Clément and Mitidieri [49] and Clément and Pr¨ uss [54]), the idea in Gripenberg [153] was to approximate L by means of its Yosida approximation, rather than A. This has the advantage that much more structural information is available for Lμ than for Aμ , since L is linear, m-accretive, and the semigroups generated by −L are positive in any space Lp (J; X) (w.r.t. the standard cone generated by a cone P in X), where J = [0, T ] or J = ⺢+ . The paper Gripenberg [153] contains the most comprehensive treatment of (13.67), hence also of (13.61). He does not need to assume k0 > 0 nor is k1 (0+) < ∞ required, in particular a(0+) = ∞ is admitted. By means of the inequality (13.66) the perturbation argument involving the contraction mapping principle allows also nonlinear functional right hand sides G(u)(t) instead of g(t) in (13.67). Thus these results extend those of Crandall and Nohel [62] considerably. Another recent approach taken in Egberts [108] (see also Clément and Egberts [46]) follows the route taken in Section 8 for the linear case, i.e. uses the sum approach writing (13.67) in abstract form as Lu + Au = h. This approach yields better results if X is a Hilbert space and either A = ∂φ and k(t) is a creep function, or if A is m-accretive and k0 = 0, | arg k1 (λ)| ≤ θ ≤ π/2 on ⺓+ , i.e. if a(t) is θ-sectorial; cp. Section 8. There are many papers that consider equations which generalize (13.61) in one way or another, and it would be too much here to list all references. We only mention the following. In the papers of Barbu and Malik [19], Crandall, Londen, and Nohel [61], Engler [110], Londen and Nohel [220], Staffans [312], the problem

t

u(t) ˙ + Bu(t) + 0

a(t − s)Au(s)ds f (t), u(0) = x,

t ≥ 0,

(13.68)


345

is studied in a Hilbert space X, where A and B are both subdifferentials of proper l.s.c. convex functionals which are related by an angle condition and subject to some compactness assumptions. Perturbation results for (13.61) of various kinds have been obtained in Aizicovici [3], [5], [6], Aizicovici, Londen, and Reich [7], Gripenberg [151], [149], Kiffe [195], Mitidieri [251], Mitidieri and Tosques [252]. The kernel a(t) is allowed to take values in B(X), X a Hilbert space, in the papers McCamy [230], McCamy and Mizel [231], McCamy and Wong [232], and Staffans [312], [313]. For references concerning concrete partial integro-differential problems of the forms (13.61) or (13.68) with A and B nonlinear partial differential operators and X being an appropriate function space we refer to the monograph Renardy, Hrusa, and Nohel [289].

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Index AAP (X), 294 AAP + (X), 294 AP (X), 293 BM (X), 284 BM 0 (X), 285 BV (J ; X), 2 BV 0 (J ; X), 3 Bpα,q (J ; X), 195 C(M ; X), 1 C0+ (X), 293 Cl+ (X), 294 C0α (J ; X), 62 C0∞ (Ω; X), 2 ¯ X), 2 C m (Ω; ¯ X), 2 Cbm (Ω; m ¯ X) , 2 Cub (Ω; Cb (M ; X), 1 Cs (J ; B(Z; X)), 15 Cub (M ; X), 1 Lp (Ω, Σ, μ; X), 2 Lp (Ω; X), 2 Lip(M ; X), 2 Lip0 (⺢+ ; X), 170 Pω (X), 295 W (X), 295 W + (X), 296 W0 (X), 295 W0+ (X), 296 XA , 1 Λ(X), 286 α(ρ, f ), 293 exp(f ), 293 ρ(A), 1 σ(A), 1 σ(f ), 19 r(t; μ), 36 s(t; μ), 36 w(t; τ ), 106 AM(α, β), 91 BF, 91 BIP(X), 214 B(X, Y ), 1 CBF, 94 CF, 94 CM, 90

D(A), 1 H(X), 284 H∞ (X), 285 Hσ (X), 286 Ha (X), 285 Hb (X), 286 Hm (X), 285 HΛ (X), 286 N (A), 1 R(A), 1 S(X), 212 W(⺢), 286 (A1),(A2),(A3), 52 (AN1),(AN2),(AN3), 186 (AN4), 187 (F), 276 (G1),(G2), 297 (G3),(G4), 298 (G5), 299 (H1),(H2), 44 (H3),(H4), 48 (I1),(I2),(I3), 262 (IR1),(IR2),(IR3), 282 (K1),(K2), 47 (N1’),(N2’), 166 (N1),(N2),(N3), 163 (NP1),(NP2),(NP3), 188 (NP4), 190 (P1),(P2), 68 (PV1),(PV2),(PV3), 191 (PV4), 192 (R1),(R2),(R3), 46 (S), 275 (S1),(S2),(S3), 32 (S1),(S2),(S3),(S4), 152 (SL1),(SL2), 212 (V1),(V2),(V3), 179 (V4),(V5), 274 (WA1),(WA2),(WA3), 186 asymptotic equivalence, 300 Banach space ζ-convex, 217 UMD-property, 233 of class HT , 217

J. Prüss, Evolutionary Integral Equations and Applications, DOI 10.1007/978-3-0348-0499-8, © Springer Basel 1993

363

364 beam, 236 Euler-Bernoulli, 239 mean angle of rotation, 237 mean displacement, 237 Rayleigh, 238 Timoshenko, 238 vicoelastic, 236 Bernstein function, 91 complete, 94 theorem, 90 Cauchy problem, abstract, 30 conductivity equilibrium, 133 instantaneous, 133 constitutive relations, 249 convolution, 3 cosine family, 33, 40 electromagnetic dispersion, 249 energy equality, 123 equation Lp -stable (R), 256 Lp -stable (S), 256 complex spectrum, 297 first resolvent, 153 integral resolvent, 46 nonautonomous, 184 parabolic, 68, 188 real spectrum, 289 resolvent, 32 second resolvent, 153 strongly Lp -stable, 229 subordinate, 101 variational, 176, 191, 274, 280 weakly parabolic, 188 well-posed, 31 with main part, 157 estimate L∞ -, 200 fluid, 130 ideally viscous, 125 Maxwell, 125 Newtonian, 130 formula complex inversion, 4 variation of parameters, 14, 33, 155 Fourier multiplier, 215 symbol, 215 transform, 3 Fourier-Carleman transform, 3 fractional derivative model, 132 function

Index 1-regular, 221 θ-sectorial, 221 k-monotone, 72, 83 k-regular, 69 p-standard, 116 absolutely monotonic, 91 almost periodic, 293 asymptotically almost periodic, 293 Bernstein, 91, 115 bounded in mean, 284 complete Bernstein, 94 completely monotonic, 55, 90 completely positive, 96, 116 creep, 94, 128 locally analytic, 260 of exponential growth, 4 of polynomial growth, 22 propagation, 106 spectrum, 19 support, 2 weakly almost periodic, 295 weakly asympt. almost periodic, 296 function space homogeneous, 284 b-admissible, 287 functional calculus of type H ∞ (Ω), 233 Hardy inequality, 81, 265 space, 81 heat capacity equilibrium, 133 instantaneous, 133 Helmholtz decomposition, 142 Hookean solid, 130 inequality dissipation, 127, 247 energy, 171, 176, 182 Hardy, 81, 265 Korn, 243, 248 Poincar´ e, 243, 248 strong dissipation, 247 interpolation space complex, 214 real, 196 kernel Λ-, 291 θ-sectorial, 68 k-regular, 188 completely monotonic, 127 of positive type, 171 positive type, 127 potential, 116

Index sectorial, 68 weakly k-regular, 188 Korn’s inequality, 243 Laplace operator, 139, 149 transform, 3 Marcienkiewicz multiplier theorem, 215 vector-valued, 217 material almost separable, 245 compressible, 123 homogeneous, 125 incompressible, 123 isotropic, 125 power type, 131 relaxation type, 133 rigid, 129 separable, 244, 245 synchronous, 126, 138 viscoelastic, 124 viscous, 129 maximal regularity, 217, 318 of type Bpα,q , 196 of type C α , 62, 194 of type Lp , 227 Maxwell’s equations, 248 mean relaxation time, 130 mean retardation time, 130 measure completely positive, 96 Laplace transformable, 4 of exponential growth, 4 of positive type, 38 Mikhlin multiplier condition, 215 mixed derivative theorem, 234 modulus compression, 126 creep, 124 elasticity, 130 shear, 126 stress relaxation, 124 tensile, 138 operator bounded imaginary powers, 214 causal, 223 coercive pair, 234 commuting, 218 complex powers, 212 elasticity, 144, 150 Laplace, 139, 149 negative semidefinite, 38 normal, 37 sectorial, 56, 212

365 power angle, 214 spectral angle, 218 selfadjoint, 38 spectrum, 1 Stokes, 142, 150, 250 thermoelasticity, 145, 150 operator family integrable, 256 strongly continuous, 15 strongly integrable, 256 uniformly integrable, 256 Paley-Wiener lemma, 16 parabolicity condition, 218 strong, 218 plate, 240 Kirchhoff, 241 Midlin-Timoshenko, 241 viscoelastic, 240 Poincar´ e’s inequality, 243 pressure, 123 Radon-Nikodym property, 154 Rayleigh beam, 238 problem, 135 relation constitutive, 124 stress-strain, 124 relaxation function energy-temperature, 133 heat conduction, 133 relaxation time, 132 resolvent, 32, 153 a-regular, 153 analytic, 49, 185 analytic weak, 185 analyticity type of, 49 differentiable, 34 distribution, 48 duality theory, 48 equation, 13, 32 exponentially bounded, 42 growth bound, 43 integral, 46 integrated, 48 kernel, 14 pseudo-, 152 type, 43 weak, 173 whole-line, 15 rigidity, 129 Schwartz space, 3 semigroup, 33, 40 analytic, 55 positive translation invariant, 116

366 sesquilinear form, 190 η-coercive, 181 coercive, 176 locally integrable, 190 uniformly measurable, 190 shear correction coefficient, 236 simple shear, 135 simple tension, 138 solid, 130 ideally elastic, 124 Kelvin-Voigt, 125 Poynting-Thompson, 131 solution mild, 30, 152, 287 operator, 32 strong, 30, 152, 287 weak, 30 Stokes operator, 142, 150 strain, 122 deviatoric, 126 stress, 122 deviatoric, 126 extra, 123 subordination principle, 116 for measures, 99 for resolvents, 101 thermoviscoelasticity, 133 transform Bohr, 293 Fourier, 3 Fourier-Carleman, 3, 19 Hilbert, 216 Laplace, 3 translation group, 284 invariance, 284 viscoelasticity, 119 viscosity dynamic, 129 static, 130 wave front, 136 propagation, 136 speed, 136 well-posedness, 31, 157 a-regular, 158 Wiener class, 286

Index

Evolutionary Integral Equations and Applications

Evolutionary Integral Equations and Applications

Integral Equations and Applications

Integral equations and applications

Integral Equations and their Applications

Integral equations and their applications

Abel Integral Equations: Analysis and Applications

Hypersingular integral equations and their applications

Differential and integral equations

Differential and Integral Equations

Differential and Integral Equations

Integral Equations

Evolutionary Equations

Integral Representations and Applications

Integral inequalities and applications

Volterra integral and functional equations

Volterra Integral and Functional Equations

Volterra integral and differential equations

Linear and Nonlinear Integral Equations: Methods and Applications

Linear and Nonlinear Integral Equations: Methods and Applications

Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications, Volume 8)

Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications)

Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications)

Volterra Integral and Differential Equations

Multidimensional Integral Equations and Inequalities

Multidimensional Integral Equations and Inequalities

Volterra Stieltjes-integral equations

Boundary Integral Equations

Linear and Nonlinear Integral Equations: Methods and Applications

Handbook of Integral Equations

Handbook of integral equations

Evolutionary Integral Equations and Applications